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

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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...`

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

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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:

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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:

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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:

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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:

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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:

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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:

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0/3 + 1/3^{2} + 1/3^{3} + 2/3^{4} + 3/3^{5} + 5/3^{6} + 8/3^{7} + 13/3^{8} + 21/3^{9} + 34/3^{10} + 55/3^{11} + 89/3^{12} + 144/3^{13} + 233/3^{14} + 377/3^{15} + 610/3^{16} + 987/3^{17} + 1597/3^{18} + 2584/3^{19} + 4181/3^{20} +... = 1/5 = 0·012101210121012101210 in b3

0/4 + 1/4^{2} + 1/4^{3} + 2/4^{4} + 3/4^{5} + 5/4^{6} + 8/4^{7} + 13/4^{8} + 21/4^{9} + 34/4^{10} + 55/4^{11} + 89/4^{12} + 144/4^{13} + 233/4^{14} + 377/4^{15} + 610/4^{16} + 987/4^{17} + 1597/4^{18} + 2584/4^{19} + 4181/4^{20} +... = 1/11 = 0·011310113101131011310 in b4

0/5 + 1/5^{2} + 1/5^{3} + 2/5^{4} + 3/5^{5} + 5/5^{6} + 8/5^{7} + 13/5^{8} + 21/5^{9} + 34/5^{10} + 55/5^{11} + 89/5^{12} + 144/5^{13} + 233/5^{14} + 377/5^{15} + 610/5^{16} + 987/5^{17} + 1597/5^{18} + 2584/5^{19} + 4181/5^{20} +... = 1/19 = 0·011242141011242141011 in b5

0/6 + 1/6^{2} + 1/6^{3} + 2/6^{4} + 3/6^{5} + 5/6^{6} + 8/6^{7} + 13/6^{8} + 21/6^{9} + 34/6^{10} + 55/6^{11} + 89/6^{12} + 144/6^{13} + 233/6^{14} + 377/6^{15} + 610/6^{16} + 987/6^{17} + 1597/6^{18} + 2584/6^{19} + 4181/6^{20} +... = 1/29 = 0·011240454431510112404 in b6

0/7 + 1/7^{2} + 1/7^{3} + 2/7^{4} + 3/7^{5} + 5/7^{6} + 8/7^{7} + 13/7^{8} + 21/7^{9} + 34/7^{10} + 55/7^{11} + 89/7^{12} + 144/7^{13} + 233/7^{14} + 377/7^{15} + 610/7^{16} + 987/7^{17} + 1597/7^{18} + 2584/7^{19} + 4181/7^{20} +... = 1/41 = 0·011236326213520225056 in b7

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

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0/8 + 1/8`^{2} + 1/8^{3} + 2/8^{4} + 3/8^{5} + 5/8^{6} + 8/8^{7} + 13/8^{8} + 21/8^{9} + 34/8^{10} + 55/8^{11} + 89/8^{12} + 144/8^{13} + 233/8^{14} + 377/8^{15} + 610/8^{16} + 987/8^{17} + 1597/8^{18} + 2584/8^{19} + 4181/8^{20} +... = 1/55 = 0·011236202247440451710 in b8

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

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0/9 + 1/9^{2} + 1/9^{3} + 2/9^{4} + 3/9^{5} + 5/9^{6} + 8/9^{7} + 13/9^{8} + 21/9^{9} + 34/9^{10} + 55/9^{11} + 89/9^{12} + 144/9^{13} + 233/9^{14} + 377/9^{15} + 610/9^{16} + 987/9^{17} + 1597/9^{18} + 2584/9^{19} + 4181/9^{20} +... = 1/71 (another prime) = 0·011236067540450563033 in b9

0/10 + 1/10^{2} + 1/10^{3} + 2/10^{4} + 3/10^{5} + 5/10^{6} + 8/10^{7} + 13/10^{8} + 21/10^{9} + 34/10^{10} + 55/10^{11} + 89/10^{12} + 144/10^{13} + 233/10^{14} + 377/10^{15} + 610/10^{16} + 987/10^{17} + 1597/10^{18} + 2584/10^{19} + 4181/10^{20} +... = 1/89 (and another) = 0·011235955056179775280 in b10

0/11 + 1/11^{2} + 1/11^{3} + 2/11^{4} + 3/11^{5} + 5/11^{6} + 8/11^{7} + 13/11^{8} + 21/11^{9} + 34/11^{10} + 55/11^{11} + 89/11^{12} + 144/11^{13} + 233/11^{14} + 377/11^{15} + 610/11^{16} + 987/11^{17} + 1597/11^{18} + 2584/11^{19} + 4181/11^{20} +... = 1/109 (and another) = 0·011235942695392022470 in b11

0/12 + 1/12^{2} + 1/12^{3} + 2/12^{4} + 3/12^{5} + 5/12^{6} + 8/12^{7} + 13/12^{8} + 21/12^{9} + 34/12^{10} + 55/12^{11} + 89/12^{12} + 144/12^{13} + 233/12^{14} + 377/12^{15} + 610/12^{16} + 987/12^{17} + 1597/12^{18} + 2584/12^{19} + 4181/12^{20} +... = 1/131 (and another) = 0·011235930336A53909A87 in b12

`0/13 + 1/13`^{2} + 1/13^{3} + 2/13^{4} + 3/13^{5} + 5/13^{6} + 8/13^{7} + 13/13^{8} + 21/13^{9} + 34/13^{10} + 55/13^{11} + 89/13^{12} + 144/13^{13} + 233/13^{14} + 377/13^{15} + 610/13^{16} + 987/13^{17} + 1597/13^{18} + 2584/13^{19} + 4181/13^{20} +... = 1/155 (not a prime or a Fibonacci number) = 0·01123591ACAA861794044 in b13

The reciprocals go like this:

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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:

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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.