Mötley Vüe

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... — A020995 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)

Wolfwords

• მელიამ მგელს შესძახა: შე უმი ხორცის ჭამიაო!
•• Meliam mgels šesdzakha: še umi khartsis ch’amio!
••• FOX-agentive WOLF-dative called: thou raw MEAT-genitive EATER-vocative
•••• The fox called to the wolf: “Thou eater of raw meat!”
••••• The pot called the kettle black.

Toxic Turntable #24

Currently listening…

• We Worship Silence, Pass the Gates (2011)
• House of Pyromania, Many Seek (Few Find) (1987)
• X-Newly Inc, Oz Wuwu 9 (2003)
• Iujisba, Abominable Abdominal (Killer Bees EP) (1986)
• Danny Yaup, Vision Ov (1969)
• Fizzy Glamsters, Keict (1991)
• Roxane Redmoor, Voxational DJ (2008)
• Kogar Fjö, Capnotic Micrographs (1993)
• Dynamic and the Zone, Cocodrilo Rock (1977)
• იჰვიუხე, პეპლები მთვარის (2002)
• Quickfinger, Ship on a Painted Ocean (1980)
• Earl Vanburgh, Glad (but) Sad (1965)
• Aquilæ ζ, Songs of Seventeen Stars (1979)
• Kozmik Krusaders, Hexen Zoo (1998)
• Quanta Thalassia, This Is Si Siht (2010)
• Thallium Addicts, Thanatographic (1999)
• X Xepj Xo, On an Ebb (2014)
• Orion’s Cradle, Live in Oslo (1987)
• Gazing on Bifrost, By the Swords (1976)
• Ausna, Z.M.E. (1977)
• Obelisk Pact, Long You’ll Slide (2003)
• Um Nuhotóbareac, L’Xac Rey (2011)


Previously pre-posted:

Toxic Turntable #1#2#3#4#5#6#7#8#9#10#11#12#13#14#15#16#17#18#19#20#21#22#23

Discharming Manc

A passionately socialist Anglican priest and proud member of the LGBTQ+ Community no longer approves of Moz:

The song I can no longer listen to

“This Charming Man”. Much as I like the song, Morrissey has ceased to be charming for me.

‘No Jacket Required would be the soundtrack of hell’: the Rev Richard Coles’s honest playlist, The Guardian, 10i22

Core War…

In terms of my core ambitions for 2022, I hope to continue the fight against such things as the reprehensible and repulsive phrase “in terms of”, the pretentious and throbbingly urgent adjective “core”, and the cheap trick of trailing dots… I know that I won’t win and that the Hive-Mind will continue to buzz deafeningly at core venues like The Guardian, The London Review of Books and The Shropshire Advertiser, but so what? In the core words of Samuel in terms of Johnson:

[I]t remains that we retard what we cannot repel, that we palliate what we cannot cure. Life may be lengthened by care, though death cannot be ultimately defeated: tongues, like governments, have a natural tendency to degeneration; we have long preserved our constitution, let us make some struggles for our language. — Samuel Johnson, Preface to a Dictionary of the English Language (1755)


Elsewhere Other-Accessible

Ex-term-in-ate! — core interrogation of why “in terms of” is so despicable, deplorable and downright disgusting…
Don’t Do Dot — core interrogation of why “…” is so despicable, deplorable and downright disgusting dot dot dot


Post-Performative Post-Scriptum

How should the first line of this incendiary intervention begin? I suggest: “In terms of my core ambitions for 2022…” → “Among my main ambitions…”

Triangular Squares

The numbers that are both square and triangular are beautifully related to the best approximations to √2:

Number

Square Root

Factors of root

1 1 1
36 6 2 * 3
1225 35 5 * 7
41616 204 12 * 17

and so on.

In each case the factors of the root are the numerator and denominator of the next approximation to √2. — David Wells, The Penguin Dictionary of Curious and Interesting Mathematics (1986), entry for “36”.


Elsewhere other-accessible

A001110 — Square triangular numbers: numbers that are both triangular and square

XXXI-Word

I enjoy doing crosswords occasionally, but I’m not very good at them. Even so, I’m still surprised at how hard I can find a kind of crossword where you look at three words and have to find another word that links them. Some of the answers can be very simple, but it sometimes takes me a long time to get them. Here’s an example with an attractively symmetric grids:

Across

1. Band, Farthing, Top
2. Jobs, Less, While
5. Bullet, Money, Surgeon
7. Back, Bank, Over
8. Half, Hiker, Up
9. Golden, Maple, Rosehip
11. Razor, Shooter, Tongue
13. Lunar, Solar, Total
14. Break, Buckets, Shirt
15. Angle, Away, Down

Down

1. Board, Roll, Sweet
2. Alec, Out, Phone
3. Night, Tower, Wrist
4. Cross, Loft, Serving
5. Dog, Oyster, Wolf
6. Cheese, Industry, Pie
9. Gum, Platform, Snow
10. Light, Test, Whale
11. Market, Power, Sonic
12. Ball, Stripper, Wet

Performativizing Papyrocentricity #69

Papyrocentric Performativity Presents:

Psyches and Psychoses – the work of Guy de Maupassant

Buzz OffThe Wasp Factory, Iain Banks (1984)

Drink InkThe Way to Dusty Death, Alistair MacLean (1973)

LittleratureIn Miniature: How Small Things Illuminate the World, Simon Garfield (Canongate 2018)

Le Paon dans les PyrénéesThe Man in the Red Coat, Julian Barnes (Penguin 2019)

Bon and OffTwo Sides to Every Glory: AC/DC: The Complete Biography, Paul Stenning (Chrome Dreams 2005)

The Fuel in the SkullThe Jewel in the Skull, Michael Moorcock (1969)

Suspicious SubstanceSubstance: Inside New Order, Peter Hook (Simon & Schuster, 2016)


Or Read a Review at Random: RaRaR