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)`