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 + 1/3

^{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:

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:

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:

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.