A213277 - OEIS

A213277

a(n) is the length of the cycle of first differences of k such that Fibonacci(k) mod n = k mod n.

3, 8, 3, 8, 6, 16, 3, 8, 14, 10, 4, 28, 24, 16, 3, 36, 6, 18, 11, 16, 15, 48, 3, 18, 42, 8, 12, 14, 14, 30, 3, 40, 18, 32, 4, 76, 9, 56, 11, 40, 12, 88, 15, 16, 24, 32, 3, 16, 34, 24, 21, 108, 6, 8, 6, 24, 21, 58, 12, 60, 15, 16, 3, 56, 30, 136, 9, 16, 56

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OFFSET

2,1

COMMENTS

In calculating the terms, a set of values A that is found twice (AA) is not enough to be certain that A is a cycle since the continuation may be AAA..AAABAAA..AAAB where B is a different set of values. In calculating the data above, a cycle is accepted when it has occurred 10 times in a row.

LINKS

Lars Blomberg, Table of n, a(n) for n = 2..23002

EXAMPLE

Example with n=3:

Fib(k): 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, ...

Fib(k) mod 3: 0,1,1,2,0,2,2,1,0,1,1,2,0,2,2,1,0,1,1,2,0,2,2,1,0

k mod 3: 0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0

Accepted k values indicated by x:

x,x,-,-,-,x,-,x,-,-,x,x,x,-,-,-,-,-,-,-,-,-,-,-,x

Accepted k values: 0, 1, 5, 7, 10, 11, 12, 14, 24

First differences of k values: 1, 4, 2, 3, 1, 1, 2, 10

After this the cycle repeats, so a(3) = 8.