A Walk on the Wide Side

Friday, 23rd Jun, 2023 by Krilling for Company in Base Behavior, Digit-sums, Fibonacci Sequence, Integer Sequences, Irrational numbers, φ, Mathematics, Phi and tagged Arithmetic, base 10, base 11, base 12, carries, carry, digit-sums of Fibonacci numbers, duodecimal, logarithms, powers of 2, recreational math, recreational mathematics, recreational maths, undecimal, undenary, unodecimal | Leave a comment

How wide is a number? The obvious answer is to count digits and say that 1 and 9 are one digit wide, 11 and 99 are two digits wide, 111 and 999 are three digits wide, and so on. But that isn’t a very good answer. 111 and 999 are both three digits wide, but 999 is nine larger times than 111. And although 111 and 999 are both one digit wider than 11 and 99, 111 is much closer to 99 than 999 is to 111.

So there’s got to be a better answer to the question. I came across it indirectly, when I started looking at carries in powers. I wanted to know how fast a number grew in digit-width as it was multiplied repeatedly by, say, 2. For example, 2^3 = 8 and 2^4 = 16, so there’s been a carry at the far left and 2^4 = 16 has increased in digit-width by 1 over 2^3 = 8. After that, 2^6 = 64 and 2^7 = 128, so there’s another carry and another increase in digit-width. I wrote a program to sum the carries and divide them by the power. If I were better at math, I would’ve known what the value of carries / power was going to be. Here’s the program beginning to find it (it begins with a carry of 1, to mark 2^0 = 1 as creating a digit ex nihilo, as it were):

8 = 2^3 16 = 2^4 → 2 / 4 = 0.5 64 = 2^6 128 = 2^7 → 3 / 7 = 0.4285714285714285714285714286 512 = 2^9 1024 = 2^10 → 4 / 10 = 0.4 8192 = 2^13 16384 = 2^14 → 5 / 14 = 0.3571428571428571428571428571 65536 = 2^16 131072 = 2^17 → 6 / 17 = 0.3529411764705882352941176471 524288 = 2^19 1048576 = 2^20 → 7 / 20 = 0.35 8388608 = 2^23 16777216 = 2^24 → 8 / 24 = 0.3... 67108864 = 2^26 134217728 = 2^27 → 9 / 27 = 0.3... 536870912 = 2^29 1073741824 = 2^30 → 10 / 30 = 0.3... 8589934592 = 2^33 17179869184 = 2^34 → 11 / 34 = 0.3235294117647058823529411765 68719476736 = 2^36 137438953472 = 2^37 → 12 / 37 = 0.3243243243243243243243243243 549755813888 = 2^39 1099511627776 = 2^40 → 13 / 40 = 0.325 8796093022208 = 2^43 17592186044416 = 2^44 → 14 / 44 = 0.318... 70368744177664 = 2^46 140737488355328 = 2^47 → 15 / 47 = 0.3191489361702127659574468085 562949953421312 = 2^49 1125899906842624 = 2^50 → 16 / 50 = 0.32 9007199254740992 = 2^53 18014398509481984 = 2^54 → 17 / 54 = 0.3148... 72057594037927936 = 2^56 144115188075855872 = 2^57 → 18 / 57 = 0.3157894736842105263157894737 576460752303423488 = 2^59 1152921504606846976 = 2^60 → 19 / 60 = 0.316... 9223372036854775808 = 2^63 18446744073709551616 = 2^64 → 20 / 64 = 0.3125 73786976294838206464 = 2^66 147573952589676412928 = 2^67 → 21 / 67 = 0.3134328358208955223880597015 590295810358705651712 = 2^69 1180591620717411303424 = 2^70 → 22 / 70 = 0.3142857... 9444732965739290427392 = 2^73 18889465931478580854784 = 2^74 → 23 / 74 = 0.3108... 75557863725914323419136 = 2^76 151115727451828646838272 = 2^77 → 24 / 77 = 0.3116883... 604462909807314587353088 = 2^79 1208925819614629174706176 = 2^80 → 25 / 80 = 0.3125 9671406556917033397649408 = 2^83 19342813113834066795298816 = 2^84 → 26 / 84 = 0.3095238095238095238095238095 77371252455336267181195264 = 2^86 154742504910672534362390528 = 2^87 → 27 / 87 = 0.3103448275862068965517241379 618970019642690137449562112 = 2^89 1237940039285380274899124224 = 2^90 → 28 / 90 = 0.31... 9903520314283042199192993792 = 2^93 19807040628566084398385987584 = 2^94 → 29 / 94 = 0.3085106382978723404255319149 79228162514264337593543950336 = 2^96 158456325028528675187087900672 = 2^97 → 30 / 97 = 0.3092783505154639175257731959 633825300114114700748351602688 = 2^99 1267650600228229401496703205376 = 2^100 → 31 / 100 = 0.31

After calculating 2^p higher and higher (I discarded trailing digits of 2^p), I realized that the answer — carries / power — was converging on a value of slightly less than 0.30103. In the end (doh!), I realized that what I was calculating was the logarithm of 2 in base 10:

log(2) = 0.3010299956639811952137388947... 10^0.301029995663981... = 2

You can use then same carries-and-powers method to approximate the values of other logarithms:

log(1) = 0 log(2) = 0.3010299956639811952137388947... log(3) = 0.4771212547196624372950279033... log(4) = 0.6020599913279623904274777894... log(5) = 0.6989700043360188047862611053... log(6) = 0.7781512503836436325087667980... log(7) = 0.8450980400142568307122162586... log(8) = 0.9030899869919435856412166842... log(9) = 0.9542425094393248745900558065...

I also realized logarithms are a good answer to the question I raised above: How wide is a number? The logs of the powers of 2 are multiples of log(2):

log(2^1) = log(2) = 0.301029995663981195213738894 log(2^2) = log(4) = 0.602059991327962390427477789 = 2 * log(2) log(2^3) = log(8) = 0.903089986991943585641216684 = 3 * log(2) log(2^4) = log(16) = 1.204119982655924780854955579 = 4 * log(2) log(2^5) = log(32) = 1.505149978319905976068694474 = 5 * log(2) log(2^6) = log(64) = 1.806179973983887171282433368 = 6 * log(2) log(2^7) = log(128) = 2.107209969647868366496172263 = 7 * log(2) log(2^8) = log(256) = 2.408239965311849561709911158 = 8 * log(2) log(2^9) = log(512) = 2.709269960975830756923650053 = 9 * log(2) log(2^10) = log(1024) = 3.010299956639811952137388947 = 10 * log(2)

4 is 2 times larger than 2 and, in a sense, the width of 4 is 0.301029995663981… greater than the width of 2. As you can see, when the integer part of the log-sum increases by 1, so does the digit-width of the power:

log(2^3) = log(8) = 0.903089986991943585641216684 = 3 * log(2) log(2^4) = log(16) = 1.204119982655924780854955579 = 4 * log(2)
[...] log(2^6) = log(64) = 1.806179973983887171282433368 = 6 * log(2) log(2^7) = log(128) = 2.107209969647868366496172263 = 7 * log(2) [...]
log(2^9) = log(512) = 2.709269960975830756923650053 = 9 * log(2) log(2^10) = log(1024) = 3.01029995663981195213738894 = 10 * log(2)

In other words, powers of 2 are increasing in width by 0.301029995663981… units. When the increase flips the integer part of the log-sum up by 1, the digit-width or digit-count also increases by 1. To find the digit-count of a number, n, in a particular base, you simply take the integer part of log(n,b) and add 1. In base 10, the log of 123456789 is 8.091514… The integer part is 8 and 8+1 = 9. But it also makes perfect sense that log(1) = 0. No matter how many times you multiply a number by 1, the number never changes. That is, its width stays the same. So you can say that 1 has a width of 0, while 2 has a width of 0.301029995663981…

Logarithms also answer a question pre-previously raised on Overlord of the Über-Feral: Why are the Fibonacci numbers so productive in base 11 for digsum(fib(k)) = k? In base 10, such numbers are quickly exhausted:

digsum(fib(1)) = 1 = digsum(1) digsum(fib(5)) = 5 = digsum(5) digsum(fib(10)) = 10 = digsum(55) digsum(fib(31)) = 31 = digsum(1346269) digsum(fib(35)) = 35 = digsum(9227465) digsum(fib(62)) = 62 = digsum(4052739537881) digsum(fib(72)) = 72 = digsum(498454011879264) digsum(fib(175)) = 175 = digsum(1672445759041379840132227567949787325) digsum(fib(180)) = 180 = digsum(18547707689471986212190138521399707760) digsum(fib(216)) = 216 = digsum(619220451666590135228675387863297874269396512) digsum(fib(251)) = 251 = digsum(12776523572924732586037033894655031898659556447352249) digsum(fib(252)) = 252 = digsum(20672849399056463095319772838289364792345825123228624) digsum(fib(360)) = 360 digsum(fib(494)) = 494 digsum(fib(540)) = 540 digsum(fib(946)) = 946 digsum(fib(1188)) = 1188 digsum(fib(2222)) = 2222

In base 11, such numbers go on and on:

digsum(fib(1),b=11) = 1 = digsum(1) (k=1) digsum(fib(5),b=11) = 5 = digsum(5) (k=5) digsum(fib(12),b=11) = 12 = digsum(1A2) (k=13) digsum(fib(38),b=11) = 38 = digsum(855138A1) (k=41) digsum(fib(49)) = 49 = digsum(2067A724762) (k=53) (c=5) digsum(fib(50)) = 50 = digsum(542194A6905) (k=55) digsum(fib(55)) = 55 = digsum(54756364A280) (k=60) digsum(fib(56)) = 56 = digsum(886283256841) (k=61) digsum(fib(82)) = 82 = digsum(57751318A9814A6410) (k=90) digsum(fib(89)) = 89 = digsum(140492673676A06482A2) (k=97) digsum(fib(144)) = 144 = digsum(401631365A48A784A09392136653457871) (k=169) (c=10) digsum(fib(159)) = 159 = digsum(67217257641069185100889658A1AA72A0805) (k=185) digsum(fib(166)) = 166 = digsum(26466A3A88237918577363A2390343388205432) (k=193) digsum(fib(186)) = 186 = digsum(6A963147A9599623A20A05390315140A21992A96005) (k=215) digsum(fib(221)) = 221 (k=265) (c=15) digsum(fib(225)) = 225 (k=269) digsum(fib(2A1)) = 2A1 (k=353) digsum(fib(2A3)) = 2A3 (k=355)
[...] digsum(fib(39409)) = 39409 (k=56395) digsum(fib(3958A)) = 3958A (k=56605) (c=295) digsum(fib(3965A)) = 3965A (k=56693) digsum(fib(3A106)) = 3A106 (k=57360) digsum(fib(3AA46)) = 3AA46 (k=58493) digsum(fib(40140)) = 40140 (k=58729) digsum(fib(4222A)) = 4222A (k=61500) (c=300) digsum(fib(42609)) = 42609 (k=61961) digsum(fib(42775)) = 42775 (k=62155) digsum(fib(4287A)) = 4287A (k=62281) digsum(fib(430A2)) = 430A2 (k=62669) digsum(fib(43499)) = 43499 (k=63149) (c=305) digsum(fib(435A9)) = 435A9 (k=63281) [...] digsum(fib(157476)) = 157476 (k=244140) (c=525) digsum(fib(158470)) = 158470 (k=245465) digsum(fib(159037)) = 159037 (k=246275) digsum(fib(159285)) = 159285 (k=246570) digsum(fib(159978)) = 159978 (k=247409) digsum(fib(162993)) = 162993 (k=252750) (c=530) digsum(fib(163A32)) = 163A32 (k=254135) digsum(fib(164918)) = 164918 (k=255329) digsum(fib(166985)) = 166985 (k=258065) digsum(fib(167234)) = 167234 (k=258493) digsum(fib(167371)) = 167371 (k=258655) (c=535) digsum(fib(1676A5)) = 1676A5 (k=259055) digsum(fib(16992A)) = 16992A (k=261997) [...]

When do these numbers run out in base 11? I don’t know, but I do know why there are so many of them. The answer involves the logarithm of a special number. The most famous aspect of Fibonacci numbers is that the ratio, fib(k) / fib(k-1), of successive numbers converges on an irrational constant known as Φ. Here are the first Fibonacci numbers, where fib(k) = fib(k-2) + fib(k-1) (in other words, 1+1 = 2, 1+2 = 3, 2+3 = 5, and so on):

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, ...

And here are the first ratios:

1 / 1 = 1 2 / 1 = 2 3 / 2 = 1.5 5 / 3 = 1.6... 8 / 5 = 1.6 13 / 8 = 1.625 21 / 13 = 1.6153846... 34 / 21 = 1.619047... 55 / 34 = 1.617647058823529411764705882 89 / 55 = 1.618... 144 / 89 = 1.617977528089887640449438202 233 / 144 = 1.61805... 377 / 233 = 1.618025751072961373390557940 610 / 377 = 1.618037135278514588859416446 987 / 610 = 1.618032786885245901639344262 1597 / 987 = 1.618034447821681864235055724 2584 / 1597 = 1.618033813400125234815278648 4181 / 2584 = 1.618034055727554179566563468 6765 / 4181 = 1.618033963166706529538387946 [...]

The ratios get closer and closer to Φ = 1.618033988749894848204586834… = (√5 + 1) / 2. In other words, fib(k) ≈ fib(k-1) * Φ = fib(k-1) * 1.618… in base 10. This means that the digit-length of fib(k) ≈ integer(k * log(&Phi)) + 1. In base b, the average value of a digit in a Fibonacci number is (b^2-b) / 2b. Therefore in base 10, the average value of a digit is (10^2-10) / 20 = 90 / 20 = 4.5. The average value of digsum(fib(k)) ≈ 4.5 * log(&Phi) * k = 4.5 * 0.20898764… * k = 0.940444… * k. It isn’t surprising that as fib(k) gets larger, digsum(fib(k)) tends to get smaller than k.

In base 10, anyway. But what about base 11? In base 11, log(Φ) = 0.20068091818623… and the average value of a base-11 digit in fib(k) is 5 = 110 / 22 = (11^2 – 11) / 22. Therefore the average value of digsum(fib(k)) in base 11 is 5 * log(&Phi) * k = 5 * 0.20068091818623… * k = 1.00340459… * k. The average value of digsum(fib(k)) is much closer to k and it’s not surprising that for so many fib(k) in base 11, digsum(fib(k)) = k. In base 11, log(Φ) ≈ 1/5 and because the average digval is 5, digsum(fib(k)) ≈ 5 * 1/5 * k = 1 * k = k. As we’ve seen, that isn’t true in base 10. Nor is it true in base 12, where log(Φ) = 0.1936538843826… and average digval is 5.5 = (12^2 – 12) / 24 = 132 / 24. Therefore the average value in base 12 of digsum(fib(k)) = 1.0650963641… * k. The function digsum(fib(k)) = k rapidly dries up in base 12, just as it does in base 10:

digsum(fib(1),b=12) = 1 = digsum(1) (k=1) digsum(fib(5),b=12) = 5 = digsum(5) (k=5) digsum(fib(11) = 11 = digsum(175) (k=13) digsum(fib(12) = 12 = digsum(275) (k=14) digsum(fib(75) = 75 = digsum(976446538A0863811) (k=89) (c=5) digsum(fib(80) = 80 = digsum(1B3643B50939808B400) (k=96) digsum(fib(A3) = A3 = digsum(35147A566682BB9529034402) (k=123) digsum(fib(165) = 165 (k=221) digsum(fib(283) = 283 (k=387) digsum(fib(2AB) = 2AB (k=419) (c=10) digsum(fib(39A) = 39A (k=550) digsum(fib(460) = 460 (k=648) digsum(fib(525) = 525 (k=749) digsum(fib(602) = 602 (k=866) digsum(fib(624) = 624 (k=892) (c=15) digsum(fib(781) = 781 (k=1105) digsum(fib(1219) = 1219 (k=2037)

Previously Pre-Posted…

• Mötley Vüe — more on digsum(fib(k)) = k

Two be Continued…

Saturday, 17th Jun, 2023 by Krilling for Company in Base Behavior, Digit-sums, Fibonacci Sequence, Integer Sequences, Mathematics, Phi and tagged base 10, base 11, digit-sums, Fibonacci index, Fibonacci numbers, Integer Sequences, recreational math, recreational mathematics, recreational maths, undecimal, undenary, unodecimal | Leave a comment

Here’s a useless fact that nobody interested in mathematics would ever forget: digsum(fib(2222)) = 2222. That is, if you add the digits of the 2222nd Fibonacci number, you get 2222:

fib(2222) = 104,966,721,620,282,584,734,867,037,988,863,914,269,721,309,244,628,258,918,225,835,217,264,239,539,186,480,867,849,267,122,885,365,019,934,494,625,410,255,045,832,359,715,759,649,385,824,745,506,982,513,773,397,742,803,445,080,995,617,047,976,796,168,678,756,479,470,761,439,513,575,962,955,568,645,505,845,492,393,360,201,582,183,610,207,447,528,637,825,187,188,815,786,270,477,935,419,631,184,553,635,981,047,057,037,341,800,837,414,913,595,584,426,355,208,257,232,868,908,837,817,478,483,039,310,790,967,631,454,123,105,472,742,221,897,397,857,677,674,619,381,961,429,837,434,434,636,098,678,708,225,493,682,469,561
2222 = 1 + 0 + 4 + 9 + 6 + 6 + 7 + 2 + 1 + 6 + 2 + 0 + 2 + 8 + 2 + 5 + 8 + 4 + 7 + 3 + 4 + 8 + 6 + 7 + 0 + 3 + 7 + 9 + 8 + 8 + 8 + 6 + 3 + 9 + 1 + 4 + 2 + 6 + 9 + 7 + 2 + 1 + 3 + 0 + 9 + 2 + 4 + 4 + 6 + 2 + 8 + 2 + 5 + 8 + 9 + 1 + 8 + 2 + 2 + 5 + 8 + 3 + 5 + 2 + 1 + 7 + 2 + 6 + 4 + 2 + 3 + 9 + 5 + 3 + 9 + 1 + 8 + 6 + 4 + 8 + 0 + 8 + 6 + 7 + 8 + 4 + 9 + 2 + 6 + 7 + 1 + 2 + 2 + 8 + 8 + 5 + 3 + 6 + 5 + 0 + 1 + 9 + 9 + 3 + 4 + 4 + 9 + 4 + 6 + 2 + 5 + 4 + 1 + 0 + 2 + 5 + 5 + 0 + 4 + 5 + 8 + 3 + 2 + 3 + 5 + 9 + 7 + 1 + 5 + 7 + 5 + 9 + 6 + 4 + 9 + 3 + 8 + 5 + 8 + 2 + 4 + 7 + 4 + 5 + 5 + 0 + 6 + 9 + 8 + 2 + 5 + 1 + 3 + 7 + 7 + 3 + 3 + 9 + 7 + 7 + 4 + 2 + 8 + 0 + 3 + 4 + 4 + 5 + 0 + 8 + 0 + 9 + 9 + 5 + 6 + 1 + 7 + 0 + 4 + 7 + 9 + 7 + 6 + 7 + 9 + 6 + 1 + 6 + 8 + 6 + 7 + 8 + 7 + 5 + 6 + 4 + 7 + 9 + 4 + 7 + 0 + 7 + 6 + 1 + 4 + 3 + 9 + 5 + 1 + 3 + 5 + 7 + 5 + 9 + 6 + 2 + 9 + 5 + 5 + 5 + 6 + 8 + 6 + 4 + 5 + 5 + 0 + 5 + 8 + 4 + 5 + 4 + 9 + 2 + 3 + 9 + 3 + 3 + 6 + 0 + 2 + 0 + 1 + 5 + 8 + 2 + 1 + 8 + 3 + 6 + 1 + 0 + 2 + 0 + 7 + 4 + 4 + 7 + 5 + 2 + 8 + 6 + 3 + 7 + 8 + 2 + 5 + 1 + 8 + 7 + 1 + 8 + 8 + 8 + 1 + 5 + 7 + 8 + 6 + 2 + 7 + 0 + 4 + 7 + 7 + 9 + 3 + 5 + 4 + 1 + 9 + 6 + 3 + 1 + 1 + 8 + 4 + 5 + 5 + 3 + 6 + 3 + 5 + 9 + 8 + 1 + 0 + 4 + 7 + 0 + 5 + 7 + 0 + 3 + 7 + 3 + 4 + 1 + 8 + 0 + 0 + 8 + 3 + 7 + 4 + 1 + 4 + 9 + 1 + 3 + 5 + 9 + 5 + 5 + 8 + 4 + 4 + 2 + 6 + 3 + 5 + 5 + 2 + 0 + 8 + 2 + 5 + 7 + 2 + 3 + 2 + 8 + 6 + 8 + 9 + 0 + 8 + 8 + 3 + 7 + 8 + 1 + 7 + 4 + 7 + 8 + 4 + 8 + 3 + 0 + 3 + 9 + 3 + 1 + 0 + 7 + 9 + 0 + 9 + 6 + 7 + 6 + 3 + 1 + 4 + 5 + 4 + 1 + 2 + 3 + 1 + 0 + 5 + 4 + 7 + 2 + 7 + 4 + 2 + 2 + 2 + 1 + 8 + 9 + 7 + 3 + 9 + 7 + 8 + 5 + 7 + 6 + 7 + 7 + 6 + 7 + 4 + 6 + 1 + 9 + 3 + 8 + 1 + 9 + 6 + 1 + 4 + 2 + 9 + 8 + 3 + 7 + 4 + 3 + 4 + 4 + 3 + 4 + 6 + 3 + 6 + 0 + 9 + 8 + 6 + 7 + 8 + 7 + 0 + 8 + 2 + 2 + 5 + 4 + 9 + 3 + 6 + 8 + 2 + 4 + 6 + 9 + 5 + 6 + 1

Numbers like this, where k = digsum(fib(k)), are rare. And 2222 is almost certainly the last of them. These are the relevant listings at the Online Encyclopedia of Integer Sequences:

0, 1, 5, 10, 31, 35, 62, 72, 175, 180, 216, 251, 252, 360, 494, 504, 540, 946, 1188, 2222 — A020995, Numbers k such that the sum of the digits of Fibonacci(k) is k.
0, 1, 5, 55, 1346269, 9227465, 4052739537881, 498454011879264, 1672445759041379840132227567949787325, 18547707689471986212190138521399707760, 619220451666590135228675387863297874269396512... — A067515, Fibonacci numbers with index = digit sum.

At least, they’re rare in base 10. What about other bases? Well, they’re rare in all other bases except one: base 11. When I looked there, I quickly found more than 450 numbers where digsum(fib(k),b=11) = k. So here’s an interesting little problem: Why is base 11 so productive? Or maybe I should say: Φ is base 11 so productive?

Reciprocal Recipes

Sunday, 23rd Oct, 2022 by Krilling for Company in Base Behavior, Fibonacci Sequence, Mathematics, Reciprocals and tagged base 10, base 2, binary, David Wells, decimals, Fibonacci Sequence, Penguin Dictionary of Curious and Interesting Numbers, reciprocals, recreational math, recreational mathematics, recreational maths | Leave a comment

Here’s a sequence. What’s the next number?

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

Here’s another sequence. What’s the next number?

0, 1, 1, 2, 3, 5, 8, 13, 21, 34...

Those aren’t trick questions, so the answers are 1 and 55, respectively. The second sequence is the famous Fibonacci sequence, where each number after [0,1] is the sum of the previous two numbers.

Now try dividing each of those sequences by powers of 2 and summing the results, like this:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/1024 + 1/2048 + 1/4096 + 1/8192 + 1/16384 + 1/32768 + 1/65536 + 1/131072 + 1/262144 + 1/524288 + 1/1048576 +... = ?
0/2 + 1/4 + 1/8 + 2/16 + 3/32 + 5/64 + 8/128 + 13/256 + 21/512 + 34/1024 + 55/2048 + 89/4096 + 144/8192 + 233/16384 + 377/32768 + 610/65536 + 987/131072 + 1597/262144 + 2584/524288 + 4181/1048576 +... = ?

What are the sums? I was surprised to learn that they’re identical:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/1024 + 1/2048 + 1/4096 + 1/8192 + 1/16384 + 1/32768 + 1/65536 + 1/131072 + 1/262144 + 1/524288 + 1/1048576 +... = 1
0/2 + 1/4 + 1/8 + 2/16 + 3/32 + 5/64 + 8/128 + 13/256 + 21/512 + 34/1024 + 55/2048 + 89/4096 + 144/8192 + 233/16384 + 377/32768 + 610/65536 + 987/131072 + 1597/262144 + 2584/524288 + 4181/1048576 +... = 1

I discovered this when I was playing with an old scientific calculator and calculated these sums:

5^2 + 2^2 = 29 5^2 + 4^2 = 41 5^2 + 6^2 = 61 5^2 + 8^2 = 89

The sums are all prime numbers. Then I idly calculated the reciprocal of 1/89:

1/89 = 0·011235955056179775...

The digits 011235… are the start of the Fibonacci sequence. It seems to go awry after that, but I remembered what David Wells had said in his wonderful Penguin Dictionary of Curious and Interesting Numbers (1986): “89 is the 11th Fibonacci number, and the period of its reciprocal is generated by the Fibonacci sequence: 1/89 = 0·11235…” He means that the Fibonacci sequence generates the digits of 1/89 like this, when you sum the columns and move carries left as necessary:

0 ↓1 ↓↓1 ↓↓↓2 ↓↓↓↓3 ↓↓↓↓↓5 ↓↓↓↓↓↓8 ↓↓↓↓↓↓13 ↓↓↓↓↓↓↓21 ↓↓↓↓↓↓↓↓34 ↓↓↓↓↓↓↓↓↓55 ↓↓↓↓↓↓↓↓↓↓89... ↓↓↓↓↓↓↓↓↓↓ 0112359550...

I tried this method of summing the Fibonacci sequence in other bases. Although it was old, the scientific calculator was crudely programmable. And it helpfully converted the sum into a final fraction once there were enough decimal digits:

0/3 + 1/3² + 1/3³ + 2/3⁴ + 3/3⁵ + 5/3⁶ + 8/3⁷ + 13/3⁸ + 21/3⁹ + 34/3¹⁰ + 55/3¹¹ + 89/3¹² + 144/3¹³ + 233/3¹⁴ + 377/3¹⁵ + 610/3¹⁶ + 987/3¹⁷ + 1597/3¹⁸ + 2584/3¹⁹ + 4181/3²⁰ +... = 1/5 = 0·012101210121012101210 in b3 0/4 + 1/4² + 1/4³ + 2/4⁴ + 3/4⁵ + 5/4⁶ + 8/4⁷ + 13/4⁸ + 21/4⁹ + 34/4¹⁰ + 55/4¹¹ + 89/4¹² + 144/4¹³ + 233/4¹⁴ + 377/4¹⁵ + 610/4¹⁶ + 987/4¹⁷ + 1597/4¹⁸ + 2584/4¹⁹ + 4181/4²⁰ +... = 1/11 = 0·011310113101131011310 in b4 0/5 + 1/5² + 1/5³ + 2/5⁴ + 3/5⁵ + 5/5⁶ + 8/5⁷ + 13/5⁸ + 21/5⁹ + 34/5¹⁰ + 55/5¹¹ + 89/5¹² + 144/5¹³ + 233/5¹⁴ + 377/5¹⁵ + 610/5¹⁶ + 987/5¹⁷ + 1597/5¹⁸ + 2584/5¹⁹ + 4181/5²⁰ +... = 1/19 = 0·011242141011242141011 in b5 0/6 + 1/6² + 1/6³ + 2/6⁴ + 3/6⁵ + 5/6⁶ + 8/6⁷ + 13/6⁸ + 21/6⁹ + 34/6¹⁰ + 55/6¹¹ + 89/6¹² + 144/6¹³ + 233/6¹⁴ + 377/6¹⁵ + 610/6¹⁶ + 987/6¹⁷ + 1597/6¹⁸ + 2584/6¹⁹ + 4181/6²⁰ +... = 1/29 = 0·011240454431510112404 in b6 0/7 + 1/7² + 1/7³ + 2/7⁴ + 3/7⁵ + 5/7⁶ + 8/7⁷ + 13/7⁸ + 21/7⁹ + 34/7¹⁰ + 55/7¹¹ + 89/7¹² + 144/7¹³ + 233/7¹⁴ + 377/7¹⁵ + 610/7¹⁶ + 987/7¹⁷ + 1597/7¹⁸ + 2584/7¹⁹ + 4181/7²⁰ +... = 1/41 = 0·011236326213520225056 in b7

It was interesting to see that all the reciprocals so far were of primes. I carried on:

0/8 + 1/8² + 1/8³ + 2/8⁴ + 3/8⁵ + 5/8⁶ + 8/8⁷ + 13/8⁸ + 21/8⁹ + 34/8¹⁰ + 55/8¹¹ + 89/8¹² + 144/8¹³ + 233/8¹⁴ + 377/8¹⁵ + 610/8¹⁶ + 987/8¹⁷ + 1597/8¹⁸ + 2584/8¹⁹ + 4181/8²⁰ +... = 1/55 = 0·011236202247440451710 in b8

Not a prime reciprocal, but a reciprocal of a Fibonacci number. Here are some more sums:

0/9 + 1/9² + 1/9³ + 2/9⁴ + 3/9⁵ + 5/9⁶ + 8/9⁷ + 13/9⁸ + 21/9⁹ + 34/9¹⁰ + 55/9¹¹ + 89/9¹² + 144/9¹³ + 233/9¹⁴ + 377/9¹⁵ + 610/9¹⁶ + 987/9¹⁷ + 1597/9¹⁸ + 2584/9¹⁹ + 4181/9²⁰ +... = 1/71 (another prime) = 0·011236067540450563033 in b9 0/10 + 1/10² + 1/10³ + 2/10⁴ + 3/10⁵ + 5/10⁶ + 8/10⁷ + 13/10⁸ + 21/10⁹ + 34/10¹⁰ + 55/10¹¹ + 89/10¹² + 144/10¹³ + 233/10¹⁴ + 377/10¹⁵ + 610/10¹⁶ + 987/10¹⁷ + 1597/10¹⁸ + 2584/10¹⁹ + 4181/10²⁰ +... = 1/89 (and another) = 0·011235955056179775280 in b10 0/11 + 1/11² + 1/11³ + 2/11⁴ + 3/11⁵ + 5/11⁶ + 8/11⁷ + 13/11⁸ + 21/11⁹ + 34/11¹⁰ + 55/11¹¹ + 89/11¹² + 144/11¹³ + 233/11¹⁴ + 377/11¹⁵ + 610/11¹⁶ + 987/11¹⁷ + 1597/11¹⁸ + 2584/11¹⁹ + 4181/11²⁰ +... = 1/109 (and another) = 0·011235942695392022470 in b11 0/12 + 1/12² + 1/12³ + 2/12⁴ + 3/12⁵ + 5/12⁶ + 8/12⁷ + 13/12⁸ + 21/12⁹ + 34/12¹⁰ + 55/12¹¹ + 89/12¹² + 144/12¹³ + 233/12¹⁴ + 377/12¹⁵ + 610/12¹⁶ + 987/12¹⁷ + 1597/12¹⁸ + 2584/12¹⁹ + 4181/12²⁰ +... = 1/131 (and another) = 0·011235930336A53909A87 in b12
0/13 + 1/13² + 1/13³ + 2/13⁴ + 3/13⁵ + 5/13⁶ + 8/13⁷ + 13/13⁸ + 21/13⁹ + 34/13¹⁰ + 55/13¹¹ + 89/13¹² + 144/13¹³ + 233/13¹⁴ + 377/13¹⁵ + 610/13¹⁶ + 987/13¹⁷ + 1597/13¹⁸ + 2584/13¹⁹ + 4181/13²⁰ +... = 1/155 (not a prime or a Fibonacci number) = 0·01123591ACAA861794044 in b13

The reciprocals go like this:

1/1, 1/5, 1/11, 1/19, 1/29, 1/41, 1/55, 1/71, 1/89, 1/109, 1/131, 1/155...

And it should be easy to see the rule that generates them:

5 = 1 + 4 11 = 5 + 6 19 = 11 + 8 29 = 19 + 10 41 = 29 + 12 55 = 41 + 14 71 = 55 + 16 89 = 17 + 18 109 = 89 + 20 131 = 109 + 22 155 = 131 + 24 [...]

But I don’t understand why the rule applies, let alone why the Fibonacci sequence generates these reciprocals in the first place.

Mötley Vüe

Sunday, 23rd Jan, 2022 by Krilling for Company in Base Behavior, Digit-sums, Fibonacci Sequence, Integer Sequences, φ, Mathematics and tagged base 10, base 11, David C. Terr, David Terr, digit-sums, digsum, Fibonacci numbers, Fibonacci Quarterly, Fibonacci Sequence, Integer Sequences, math, mathematics, maths, OEIS, Online Encyclopedia of Integer Sequences, recreational math, recreational mathematics, recreational maths, undecimal, undenary, unodecimal | Leave a comment

Here’s the Fibonacci sequence, where each term (after the first two) is created by adding the two previous numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765...

In “Fib and Let Tri”, I described how my eye was caught by 55, which is a palindrome, reading the same backwards and forwards. “Were there any other Fibonacci palindromes?” I wondered. So I looked to see. Now my eye has been caught by 55 again, but for another reason. It should be easy to spot another interesting aspect to 55 when the Fibonacci numbers are set out like this:

fib(1) = 1 fib(2) = 1 fib(3) = 2 fib(4) = 3 fib(5) = 5 fib(6) = 8 fib(7) = 13 fib(8) = 21 fib(9) = 34 fib(10) = 55 fib(11) = 89 fib(12) = 144 fib(13) = 233 fib(14) = 377 fib(15) = 610 fib(16) = 987 fib(17) = 1597 fib(18) = 2584 fib(19) = 4181 fib(20) = 6765 [...]

55 is fib(10), the 10th Fibonacci number, and 5+5 = 10. That is, digsum(fib(10)) = 10. What other Fibonacci numbers work like that? I soon found some and confirmed my answer at the Online Encyclopedia of Integer Sequences:

1, 5, 10, 31, 35, 62, 72, 175, 180, 216, 251, 252, 360, 494, 504, 540, 946, 1188, 2222 — A020995 at OEIS

And that seems to be the lot, according to the OEIS. In base 10, at least, but why stop at base 10? When I looked at base 11, the numbers of digsum(fib(k)) = k didn’t stop coming, because I couldn’t take the Fibonacci numbers very high on my computer. But the OEIS gives a much longer list, starting like this:

1, 5, 13, 41, 53, 55, 60, 61, 90, 97, 169, 185, 193, 215, 265, 269, 353, 355, 385, 397, 437, 481, 493, 617, 629, 630, 653, 713, 750, 769, 780, 889, 905, 960, 1013, 1025, 1045, 1205, 1320, 1405, 1435, 1501, 1620, 1650, 1657, 1705, 1735, 1769, 1793, 1913, 1981, 2125, 2153, 2280, 2297, 2389, 2413, 2460, 2465, 2509, 2533, 2549, 2609, 2610, 2633, 2730, 2749, 2845, 2893, 2915, 3041, 3055, 3155, 3209, 3360, 3475, 3485, 3521, 3641, 3721, 3749, 3757, 3761, 3840, 3865, 3929, 3941, 4075, 4273, 4301, 4650, 4937, 5195, 5209, 5435, 5489, 5490, 5700, 5917, 6169, 6253, 6335, 6361, 6373, 6401, 6581, 6593, 6701, 6750, 6941, 7021, 7349, 7577, 7595, 7693, 7740, 7805, 7873, 8009, 8017, 8215, 8341, 8495, 8737, 8861, 8970, 8995, 9120, 9133, 9181, 9269, 9277, 9535, 9541, 9737, 9935, 9953, 10297, 10609, 10789, 10855, 11317, 11809, 12029, 12175... — A025490 at OEIS

The list ends with 1636597 = A18666[b11] and the OEIS says that 1636597 almost certainly completes the list. According to David C. Terr’s paper “On the Sums of Fibonacci Numbers” (pdf), published in the Fibonacci Quarterly in 1996, the estimated digit-sum for the k-th Fibonacci number in base b is given by the formula (b-1)/2 * k * log(b,φ), where log(b,φ) is the logarithm in base b of the golden ratio, 1·61803398874… Terr then notes that the simplified formula (b-1)/2 * log(b,φ) gives the estimated average ratio digsum(fib(k)) / k in base b. Here are the estimates for bases 2 to 20:

b02 = 0.3471209568153086... b03 = 0.4380178794859424... b04 = 0.5206814352229629... b05 = 0.5979874356654401... b06 = 0.6714235829697111... b07 = 0.7418818776805580... b08 = 0.8099488992357201... b09 = 0.8760357589718848... b10 = 0.9404443811249043... b11 = 1.0034045909311624... b12 = 1.0650963641043091... b13 = 1.1256639207937723... b14 = 1.1852250528196852... b15 = 1.2438775226715552... b16 = 1.3017035880574074... b17 = 1.3587732842474014... b18 = 1.4151468584732730... b19 = 1.4708766105122322... b20 = 1.5260083080264088...

In base 2, you can expect digsum(fib(k)) to be much smaller than k; in base 20, you can expect digsum(fib(k)) to be much larger. But as you can see, the estimate for base 11, 1.0034045909311624…, is very nearly 1. That’s why base 11 produces so many results for digsum(fib(k)) = k, because only a slight deviation from the estimate might create a perfect ratio of 1 for digsum(fib(k)) / k, i.e. digsum(fib(k)) = k. But in the end the results run out in base 11 too, because as k gets higher and fib(k) gets bigger, the estimate becomes more and more accurate and digsum(fib(k)) > k. With lower k, digsum(fib(k)) can easily fall below k or match k. That happens in other bases, but because their estimates are further from 1, results for digsum(fib(k)) = k run out much more quickly.

To see this base behavior represented visually, I’ve created Ulam-like spirals for k using three colors: blue for digsum(fib(k)) < k, yellow for digsum(fib(k)) > k, and red for digsum(fib(k)) = k (with the green square at the center representing fib(1) = 1). As you can see below, the spiral for base 11 immediately stands out. It’s motley, not dominated by blue or yellow like the other spirals:

Spiral for digsum(fib(k)) in base 9
(blue for digsum(fib(k)) < k, yellow for digsum(fib(k)) > k, red for digsum(fib(k)) = k, green for fib(1))

Spiral for digsum(fib(k)) in base 10

Spiral for digsum(fib(k)) in base 11 — a motley view of blue, yellow and red

Spiral for digsum(fib(k)) in base 12

Spiral for digsum(fib(k)) in base 13

Finally, here are spirals at higher and higher resolution for digsum(fib(k)) = k in base 11:

digsum(fib(k)) = k in base 11 (low resolution)
(green square is fib(1))

digsum(fib(k)) = k in base 11 (x2 resolution)

digsum(fib(k)) = k in base 11 (x4)

digsum(fib(k)) = k in base 11 (x8)

digsum(fib(k)) = k in base 11 (x16)

digsum(fib(k)) = k in base 11 (x32)

digsum(fib(k)) = k in base 11 (x64)

digsum(fib(k)) = k in base 11 (x128)

digsum(fib(k)) = k in base 11 (animated)

Fib and Let Tri

Tuesday, 23rd Nov, 2021 by Krilling for Company in Base Behavior, Fibonacci Sequence, Integer Sequences, Irrational numbers, φ, Mathematics, Palindromic numbers, Phi and tagged base 10, base 2, base 3, base 817138163596, bases, Fibonacci numbers, Fibonacci palindromes, Fibonacci Sequence, Integer Sequences, φ, Lucas numbers, Lucas sequence, math, mathematics, maths, palindromes, palindromic Fibonacci numbers, palindromic numbers, phi, recreational math, recreational mathematics, recreational maths, tripal | Leave a comment

It’s a simple sequence with hidden depths:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155... — A000045 at OEIS
That’s the Fibonacci sequence, probably the most famous of all integer sequences after the integers themselves (1, 2, 3, 4, 5…) and the primes (2, 3, 5, 7, 11…). It has a very simple definition: if fib(fi) is the fi-th number in the Fibonacci sequence, then fib(fi) = fib(fi-1) + fib(fi-2). By definition, fib(1) = fib(2) = 1. After that, it’s easy to generate new numbers:
2 = fib(3) = fib(1) + fib(2) = 1 + 1 3 = fib(4) = fib(2) + fib(3) = 1 + 2 5 = fib(5) = fib(3) + fib(4) = 2 + 3 8 = fib(6) = fib(4) + fib(5) = 3 + 5 13 = fib(7) = fib(5) + fib(6) = 5 + 8 21 = fib(8) = fib(6) + fib(7) = 8 + 13 34 = fib(9) = fib(7) + fib(8) = 13 + 21 55 = fib(10) = fib(8) + fib(9) = 21 + 34 89 = fib(11) = fib(9) + fib(10) = 34 + 55 144 = fib(12) = fib(10) + fib(11) = 55 + 89 233 = fib(13) = fib(11) + fib(12) = 89 + 144 377 = fib(14) = fib(12) + fib(13) = 144 + 233 610 = fib(15) = fib(13) + fib(14) = 233 + 377 987 = fib(16) = fib(14) + fib(15) = 377 + 610 [...]
How to create the Fibonacci sequence is obvious. But it’s not obvious that fib(fi) / fib(fi-1) gives you ever-better approximations to a fascinating constant called φ, the golden ratio, which is 1.618033988749894…:
1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = 1.66666... 8/5 = 1.6 13/8 = 1.625 21/13 = 1.615384... 34/21 = 1.619047... 55/34 = 1.6176470588235294117647058823... 89/55 = 1.618181818... 144/89 = 1.617977528089887640... 233/144 = 1.6180555555... 377/233 = 1.618025751072961... 610/377 = 1.618037135278514... 987/610 = 1.618032786885245... [...]
And that’s just the start of the hidden depths in the Fibonacci sequence. I stumbled across another interesting pattern for myself a few days ago. I was looking at the sequence and one of the numbers caught my eye:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597...
55 is a palindrome, reading the same forward and backwards. I wondered whether there were any other palindromes in the sequence (apart from the trivial single-digit palindromes 1, 1, 2, 3…). I couldn’t find any more. Nor can anyone else, apparently. But that’s in base 10. Other bases are more productive. For example, in bases 2, 3 and 4, you get this:
11 in b2 = 3 101 in b2 = 5 10101 in b2 = 21

22 in b3 = 8 111 in b3 = 13 22122 in b3 = 233

11 in b4 = 5 111 in b4 = 21 202 in b4 = 34 313 in b4 = 55

I decided to concentrate on tripals, or palindromes with three digits. I started looking at bases that set records for the greatest number of tripals. And there are some interesting patterns in the digits of the tripals in these bases (when a digit > 9, the digit is represented inside square brackets — see base-29 and higher). See how quickly you can spot the patterns:
Palindromic Fibonacci numbers in base-4

111 in b4 (fib=21, fi=8)
202 in b4 (fib=34, fi=9)
313 in b4 (fib=55, fi=10)

4 = 2^2 (pal=3)

Palindromic Fibonacci numbers in base-11

121 in b11 (fib=144, fi=12)
313 in b11 (fib=377, fi=14)
505 in b11 (fib=610, fi=15)
818 in b11 (fib=987, fi=16)

11 is prime (pal=4)

Palindromic Fibonacci numbers in base-29

151 in b29 (fib=987, fi=16)
323 in b29 (fib=2584, fi=18)
818 in b29 (fib=6765, fi=20)
[13]0[13] in b29 (fib=10946, fi=21)
[21]1[21] in b29 (fib=17711, fi=22)

29 is prime (pal=5)

Palindromic Fibonacci numbers in base-76

1[13]1 in b76 (fib=6765, fi=20)
353 in b76 (fib=17711, fi=22)
828 in b76 (fib=46368, fi=24)
[21]1[21] in b76 (fib=121393, fi=26)
[34]0[34] in b76 (fib=196418, fi=27)
[55]1[55] in b76 (fib=317811, fi=28)

76 = 2^2 * 19 (pal=6)

Palindromic Fibonacci numbers in base-199

1[34]1 in b199 (fib=46368, fi=24)
3[13]3 in b199 (fib=121393, fi=26)
858 in b199 (fib=317811, fi=28)
[21]2[21] in b199 (fib=832040, fi=30)
[55]1[55] in b199 (fib=2178309, fi=32)
[89]0[89] in b199 (fib=3524578, fi=33)
[144]1[144] in b199 (fib=5702887, fi=34)

199 is prime (pal=7)

Palindromic Fibonacci numbers in base-521

1[89]1 in b521 (fib=317811, fi=28)
3[34]3 in b521 (fib=832040, fi=30)
8[13]8 in b521 (fib=2178309, fi=32)
[21]5[21] in b521 (fib=5702887, fi=34)
[55]2[55] in b521 (fib=14930352, fi=36)
[144]1[144] in b521 (fib=39088169, fi=38)
[233]0[233] in b521 (fib=63245986, fi=39)
[377]1[377] in b521 (fib=102334155, fi=40)

521 is prime (pal=8)

Palindromic Fibonacci numbers in base-1364

1[233]1 in b1364 (fib=2178309, fi=32)
3[89]3 in b1364 (fib=5702887, fi=34)
8[34]8 in b1364 (fib=14930352, fi=36)
[21][13][21] in b1364 (fib=39088169, fi=38)
[55]5[55] in b1364 (fib=102334155, fi=40)
[144]2[144] in b1364 (fib=267914296, fi=42)
[377]1[377] in b1364 (fib=701408733, fi=44)
[610]0[610] in b1364 (fib=1134903170, fi=45)
[987]1[987] in b1364 (fib=1836311903, fi=46)

1364 = 2^2 * 11 * 31 (pal=9)

Two patterns are quickly obvious. Every digit in the tripals is a Fibonacci number. And the middle digit of one Fibonacci tripal, fib(fi), becomes fib(fi-2) in the next tripal, while fib(fi), the first and last digits (which are identical), becomes fib(fi+2) in the next tripal.

But what about the bases? If you’re an expert in the Fibonacci sequence, you’ll spot the pattern at work straight away. I’m not an expert, but I spotted it in the end. Here are the first few bases setting records for the numbers of Fibonacci tripals:
4, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196...
These numbers come from the Lucas sequence, which is closely related to the Fibonacci sequence. But where fib(1) = fib(2) = 1, luc(1) = 1 and luc(2) = 3. After that, luc(li) = luc(li-2) + luc(li-1):
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196... — A000204 at OEIS
It seems that every second number from 4 in the Lucas sequence supplies a base in which 1) the number of Fibonacci tripals sets a new record; 2) every digit of the Fibonacci tripals is itself a Fibonacci number.

But can I prove that this is always true? No. And do I understand why these patterns exist? No. My simple search for palindromes in the Fibonacci sequence soon took me far out of my mathematical depth. But it’s been fun to find huge bases like this in which every digit of every Fibonacci tripal is itself a Fibonacci number:
Palindromic Fibonacci numbers in base-817138163596

1[139583862445]1 in b817138163596 (fib=781774079430987230203437, fi=116)
3[53316291173]3 in b817138163596 (fib=2046711111473984623691759, fi=118)
8[20365011074]8 in b817138163596 (fib=5358359254990966640871840, fi=120)
[21][7778742049][21] in b817138163596 (fib=14028366653498915298923761, fi=122)
[55][2971215073][55] in b817138163596 (fib=36726740705505779255899443, fi=124)
[144][1134903170][144] in b817138163596 (fib=96151855463018422468774568, fi=126)
[377][433494437][377] in b817138163596 (fib=251728825683549488150424261, fi=128)
[987][165580141][987] in b817138163596 (fib=659034621587630041982498215, fi=130)
[2584][63245986][2584] in b817138163596 (fib=1725375039079340637797070384, fi=132)
[6765][24157817][6765] in b817138163596 (fib=4517090495650391871408712937, fi=134)
[17711][9227465][17711] in b817138163596 (fib=11825896447871834976429068427, fi=136)
[46368][3524578][46368] in b817138163596 (fib=30960598847965113057878492344, fi=138)
[121393][1346269][121393] in b817138163596 (fib=81055900096023504197206408605, fi=140)
[317811][514229][317811] in b817138163596 (fib=212207101440105399533740733471, fi=142)
[832040][196418][832040] in b817138163596 (fib=555565404224292694404015791808, fi=144)
[2178309][75025][2178309] in b817138163596 (fib=1454489111232772683678306641953, fi=146)
[5702887][28657][5702887] in b817138163596 (fib=3807901929474025356630904134051, fi=148)
[14930352][10946][14930352] in b817138163596 (fib=9969216677189303386214405760200, fi=150)
[39088169][4181][39088169] in b817138163596 (fib=26099748102093884802012313146549, fi=152)
[102334155][1597][102334155] in b817138163596 (fib=68330027629092351019822533679447, fi=154)
[267914296][610][267914296] in b817138163596 (fib=178890334785183168257455287891792, fi=156)
[701408733][233][701408733] in b817138163596 (fib=468340976726457153752543329995929, fi=158)
[1836311903][89][1836311903] in b817138163596 (fib=1226132595394188293000174702095995, fi=160)
[4807526976][34][4807526976] in b817138163596 (fib=3210056809456107725247980776292056, fi=162)
[12586269025][13][12586269025] in b817138163596 (fib=8404037832974134882743767626780173, fi=164)
[32951280099]5[32951280099] in b817138163596 (fib=22002056689466296922983322104048463, fi=166)
[86267571272]2[86267571272] in b817138163596 (fib=57602132235424755886206198685365216, fi=168)
[225851433717]1[225851433717] in b817138163596 (fib=150804340016807970735635273952047185, fi=170)
[365435296162]0[365435296162] in b817138163596 (fib=244006547798191185585064349218729154, fi=171)
[591286729879]1[591286729879] in b817138163596 (fib=394810887814999156320699623170776339, fi=172)

817138163596 = 2^2 * 229 * 9349 * 95419 (pal=30)

Dig Sum Fib

Monday, 16th Mar, 2015 by Krilling for Company in Base Behavior, Digit-sums, Fibonacci Sequence, Mathematics and tagged adding digits, Arithmetic, digit sums, digits, Fibonacci Sequence, Integer Sequences, integers, math, mathematics, maths, numbers, recreational math, recreational mathematics, recreational maths | Leave a comment

The Fibonacci sequence is an infinitely rich sequence based on a very simple rule: add the previous two numbers. If the first two numbers are 1 and 1, the sequence begins like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025…

Plainly, the numbers increase for ever. The hundredth Fibonacci number is 354,224,848,179,261,915,075, for example, and the two-hundredth is 280,571,172,992,510,140,037,611,932,413,038,677,189,525. But there are variants on the Fibonacci sequence that don’t increase for ever. The standard rule is n(i) = n(i-2) + n(i-1). What if the rule becomes n(i) = digitsum(n(i-2)) + digitsum(n(i-1))? Now the sequence falls into a loop, like this:

1, 1, 2, 3, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10, 2, 3… (length=28)

But that’s in base 10. Here are the previous bases:

1, 1, 2, 2, 2… (base=2) (length=5)
1, 1, 2, 3, 3, 2, 3… (b=3) (l=7)
1, 1, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3… (b=4) (l=12)
1, 1, 2, 3, 5, 4, 5, 5, 2, 3… (b=5) (l=10)
1, 1, 2, 3, 5, 8, 8, 6, 4, 5, 9, 9, 8, 7, 5, 7, 7, 4, 6, 5, 6, 6, 2, 3… (b=6) (l=24)
1, 1, 2, 3, 5, 8, 7, 3, 4, 7, 5, 6, 11, 11, 10, 9, 7, 4, 5, 9, 8, 5, 7, 6, 7, 7, 2, 3… (b=7) (l=28)
1, 1, 2, 3, 5, 8, 6, 7, 13, 13, 12, 11, 9, 6, 8, 7, 8, 8, 2, 3… (b=8) (l=20)
1, 1, 2, 3, 5, 8, 13, 13, 10, 7, 9, 8, 9, 9, 2, 3… (b=9) (l=16)

Apart from base 2, all the bases repeat with (2, 3), which is set up in each case by (base, base) = (10, 10) in that base, equivalent to (1, 1). All bases > 2 appear to repeat with (2, 3), but I don’t understand why. The length of the sequence varies widely. Here it is in bases 29, 30 and 31:

1, 1, 2, 3, 5, 8, 13, 21, 34, 27, 33, 32, 9, 13, 22, 35, 29, 8, 9, 17, 26, 43, 41, 28, 41, 41, 26, 39, 37, 20, 29, 21, 22, 43, 37, 24, 33, 29, 6, 7, 13, 20, 33, 25, 30, 27, 29, 28, 29, 29, 2, 3… (b=29) (l=52)

1, 1, 2, 3, 5, 8, 13, 21, 34, 26, 31, 28, 30, 29, 30, 30, 2, 3 (b=30) (l=18)

1, 1, 2, 3, 5, 8, 13, 21, 34, 25, 29, 54, 53, 47, 40, 27, 37, 34, 11, 15, 26, 41, 37, 18, 25, 43, 38, 21, 29, 50, 49, 39, 28, 37, 35, 12, 17, 29, 46, 45, 31, 16, 17, 33, 20, 23, 43, 36, 19, 25, 44, 39, 23, 32, 25, 27, 52, 49, 41, 30, 41, 41, 22, 33, 25, 28, 53, 51, 44, 35, 19, 24, 43, 37, 20, 27, 47, 44, 31, 15, 16, 31, 17, 18, 35, 23, 28, 51, 49, 40, 29, 39, 38, 17, 25, 42, 37, 19, 26, 45, 41, 26, 37, 33, 10, 13, 23, 36, 29, 35, 34, 9, 13, 22, 35, 27, 32, 29, 31, 30, 31, 31, 2, 3 (b=31) (l=124)

The sequence for base 77 is short like that for base 30:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 68, 81, 73, 78, 75, 77, 76, 77, 77, 2, 3 (b=77) (l=22)

But the sequence for base 51 is this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 39, 44, 83, 77, 60, 37, 47, 84, 81, 65, 46, 61, 57, 18, 25, 43, 68, 61, 29, 40, 69, 59, 28, 37, 65, 52, 17, 19, 36, 55, 41, 46, 87, 83, 70, 53, 23, 26, 49, 75, 74, 49, 73, 72, 45, 67, 62, 29, 41, 70, 61, 31, 42, 73, 65, 38, 53, 41, 44, 85, 79, 64, 43, 57, 50, 57, 57, 14, 21, 35, 56, 41, 47, 88, 85, 73, 58, 31, 39, 70, 59, 29, 38, 67, 55, 22, 27, 49, 76, 75, 51, 26, 27, 53, 30, 33, 63, 46, 59, 55, 14, 19, 33, 52, 35, 37, 72, 59, 31, 40, 71, 61, 32, 43, 75, 68, 43, 61, 54, 15, 19, 34, 53, 37, 40, 77, 67, 44, 61, 55, 16, 21, 37, 58, 45, 53, 48, 51, 49, 50, 99, 99, 98, 97, 95, 92, 87, 79, 66, 45, 61, 56, 17, 23, 40, 63, 53, 16, 19, 35, 54, 39, 43, 82, 75, 57, 32, 39, 71, 60, 31, 41, 72, 63, 35, 48, 83, 81, 64, 45, 59, 54, 13, 17, 30, 47, 77, 74, 51, 25, 26, 51, 27, 28, 55, 33, 38, 71, 59, 30, 39, 69, 58, 27, 35, 62, 47, 59, 56, 15, 21, 36, 57, 43, 50, 93, 93, 86, 79, 65, 44, 59, 53, 12, 15, 27, 42, 69, 61, 30, 41, 71, 62, 33, 45, 78, 73, 51, 24, 25, 49, 74, 73, 47, 70, 67, 37, 54, 41, 45, 86, 81, 67, 48, 65, 63, 28, 41, 69, 60, 29, 39, 68, 57, 25, 32, 57, 39, 46, 85, 81, 66, 47, 63, 60, 23, 33, 56, 39, 45, 84, 79, 63, 42, 55, 47, 52, 49, 51, 50, 51, 51, 2, 3… (b=51) (l=304)

Breeding Bunnies

Tuesday, 19th Nov, 2013 by Krilling for Company in Aesthetics, Art, Book Reviews, Fibonacci Sequence, Irrational numbers, φ, Mathematics, Phi and tagged 1.6180339887498948482, ancient Greeks, Benford's law, Euclid, Fibonacci Sequence, fractal strings, golden section, irrational number, irrational numbers, φ, Mario Livio, math, mathematical constructs, mathematics, maths, nature, number patterns, phi, pineapples, rabbit-breeding, recreational math, recreational mathematics, recreational maths, Salvador Dalí, sunflowers, the flight of hawks, the golden ratio, The Golden Ratio: The Story of Phi the Extraordinary Number of Nature Art and Beauty, The Golden Ratio: The Story of Phi the World's Most Astonishing Number, the unreasonable effectiveness of mathematics | Leave a comment

The Golden Ratio: The Story of Phi, the Extraordinary Number of Nature, Art and Beauty, Mario Livio (Headline Review 2003)

A good short popular guide to perhaps the most interesting, and certainly the most irrational, of all numbers: the golden ratio or phi (φ), which is approximately equal to 1·6180339887498948482… Prominent in mathematics since at least the ancient Greeks and Euclid, phi is found in many places in nature too, from pineapples and sunflowers to the flight of hawks. Livio catalogues its appearances in both maths and nature, looking closely at the Fibonacci sequence and rabbit-breeding, before going on to debunk mistaken claims that phi also appears a lot in art, music and poetry. Dalí certainly used it, but da Vinci, Debussy and Virgil almost certainly didn’t. Nor, almost certainly, did the builders of the Parthenon and pyramids. Finally, he examines what has famously been called (by the physicist Eugene Wiegner) the unreasonable effectiveness of mathematics: why is this human invention so good at describing the behaviour of the Universe? Livio quotes one of the best short answers I’ve seen:

Human logic was forced on us by the physical world and is therefore consistent with it. Mathematics derives from logic. That is why mathematics is consistent with the physical world. (ch. 9, “Is God a mathematician?”, pg. 252)

It’s not hard to recommend a book that quotes everyone from Johannes Kepler and William Blake to Lewis Carroll, Christopher Marlowe and Jef Raskin, “the creator of the Macintosh computer”, whose answer is given above. Recreational mathematicians should also find lots of ideas for further investigation, from fractal strings to the fascinating number patterns governed by Benford’s law. It isn’t just human beings who look after number one: as a leading figure, 1 turns up much more often in data from the real world, and in mathematical constructs like the Fibonacci sequence, than intuition would lead you to expect. If you’d like to learn more about that and about many other aspects of mathematics, hunt down a copy of this book.

Elsewhere other-posted:

• Roses Are Golden – φ and floral homicide

Yew and Me

Thursday, 7th Mar, 2013 by Krilling for Company in A.E. Housman, Art, Biology, Book Reviews, Fibonacci Sequence, Fractals, Mathematics, Poetry and tagged A.E. Housman, Alan Mitchell, artist, aspen, aspens, Black Walnut, churches, churchyards, Common Walnut, Common Yew, conifer, conifers, David More, deciduous tree, deciduous trees, dendrophile, dendrophilia, Fibonacci Sequence, fractal structure, furniture, gunstocks, holy site, holy sites, Housman, illustration, illustrations, illustrator, Juglans nigra, Juglans regia, leaf, leaves, math, mathematics, maths, natural fractal, natural fractals, photosynthesis, pine-cone, pine-cones, poisonouns foliage, poisonous leaves, poisonous seeds, Populus tremula, sacred tree, sacred trees, symbol of immortality, symbols of immortality, Taxus baccata, The Pocket Guide to The Trees of Britain and Northern Europe, The Trees of Britain and Northern Europe, tree, tree-poem, tree-poems, tree-poetry, Trees, yew, yews | 3 Comments

The Pocket Guide to The Trees of Britain and Northern Europe, Alan Mitchell, illustrated by David More (1990)

Leafing through this book after I first bought it, I suddenly grabbed at it, because I thought one of the illustrations was real and that a leaf was about to slide off the page and drop to the floor. It was an easy mistake to make, because David More is a good artist. That isn’t surprising: good artists are often attracted to trees. I think it’s a mathemattraction. Trees are one of the clearest and commonest examples of natural fractals, or shapes that mirror themselves on smaller and smaller scales. In trees, trunks divide into branches into branchlets into twigs into twiglets, where the leaves, well distributed in space, wait to eat the sun.

When deciduous, or leaf-dropping, trees go hungry during the winter, this fractal structure is laid bare. And when you look at a bare tree, you’re looking at yourself, because humans are fractals too. Our torsos sprout arms sprout hands sprout fingers. Our veins become veinlets become capillaries. Ditto our lungs and nervous systems. We start big and get small, mirroring ourselves on smaller and smaller scales. Fractals make maximum and most efficient use of space and what’s found in me or thee is also found in a tree, both above and below ground. The roots of a tree are also fractals. But one big difference between trees and people is that trees are much freer to vary their general shape. Trees aren’t mirror-symmetrical like animals and that’s another thing that attracts human eyes and human artists. Each tree is unique, shaped by the chance of its seeding and setting, though each species has its characteristic silhouette. David More occasionally shows that bare winter silhouette, but usually draws the trees in full leaf, disposed to eat the sun. Trees can also be identified by their leaves alone and leaves too are fractals. The veins of a leaf divide and sub-divide, carrying the raw materials and the finished products of photosynthesis to and from the trunk and roots. Trees are giants that work on a microscopic scale, manufacturing themselves from photons and molecules of water and carbon dioxide.

We eat or sculpt what they manufacture, as Alan Mitchell describes in the text of this book:

The name “Walnut” comes from the Anglo-Saxon for “foreign nut” and was in use before the Norman Conquest, probably dating from Roman times. It may refer to the fruit rather than the tree but the Common Walnut, Juglans regia, has been grown in Britain for a very long time. The Romans associated their god Jupiter (Jove) with this tree, hence the Latin name juglans, “Jove’s acorn (glans) or nut”… The wood [of Black Walnut, Juglans nigra] is like that of Common Walnut and both are unsurpassed for use as gunstocks because, once seasoned and worked, neither moves at all and they withstand shock particularly well. They are also valued in furniture for their good colour and their ability to take a high polish. (entry for “Walnuts”, pg. 18)

That’s from the first and longer section, devoted to “Broadleaved Trees and Palms”; in the second section, “Conifers”, devoted to pines and their relatives, maths appears in a new form. Pine-cones embody the Fibonacci sequence, one of the most famous of all number sequences or series. Start with 1 and 1, then add the pair and go on adding pairs: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… That’s the Fibonacci sequence, named after the Italian mathematician Leonardo Fibonacci (c.1170-c.1245). And if you examine the two spirals created by the scales of a pine-cone, clockwise and counter-clockwise, you’ll find that there are, say, five spirals in one direction and eight in another, or eight and thirteen. The scales of a pineapple and petals of many flowers behave in a similar way. These patterns aren’t fractals like branches and leaves, but they’re also about distributing living matter efficiently through space. Mitchell doesn’t discuss any of this mathematics, but it’s there implicitly in the illustrations and underlies his text. Even the toxicity of the yew is ultimately mathematical, because the effect of toxins is determined by their chemical shape and its interaction with the chemicals in our bodies. Micro-geometry can be noxious. Or nourishing:

The Yews are a group of conifers, much more primitive than those which bear cones. Each berry-like fruit has a single large seed, partially enclosed in a succulent red aril which grows up around it. The seed is, like the foliage, very poisonous to people and many animals, but deer and rabbits eat the leaves without harm. Yew has extremely strong and durable wood [and the] Common Yew, Taxus baccata, is nearly immortal, resistant to almost every pest and disease of importance, and immune to stress from exposure, drought and cold. It is by a long way the longest-living tree we have and many in country churchyards are certainly much older than the churches, often thousands of years old. Since the yews pre-date the churches, the sites may have been holy sites and the yews sacred trees, possibly symbols of immortality, under which the Elders met. (entry for “Yews”, pg. 92)

This isn’t a big book, but there’s a lot to look at and read. I’d like a doubtful etymology to be true: some say “book” is related to “beech”, because beech-bark or beech-leaves were used for writing on. Bark is another way of identifying a tree and another aspect of their dendro-mathematics, in its texture, colours and patterns. But trees can please the ear as well as the eye: the dendrophile A.E. Housman (1859-1936) recorded how “…overhead the aspen heaves / Its rainy-sounding silver leaves” (A Shropshire Lad, XXVI). There’s maths there too. An Aspen sounds like rain in part because its many leaves, which tremble even in the lightest breeze, are acting like many rain-drops. That trembling is reflected in the tree’s scientific name: Populus tremula, “trembling poplar”. Housman, a Latin professor as well as an English poet, could have explained how tree-nouns in Latin are masculine in form: Alnus, Pinus, Ulmus; but feminine in gender: A. glutinosa, P. contorta, U. glabra (Common Alder, Lodgepole Pine, Wych-Elm). He also sums up why trees please in these simple and ancient words of English:

Give me a land of boughs in leaf,
A land of trees that stand;
Where trees are fallen, there is grief;
I love no leafless land.

More Poems, VIII.

Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

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