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%I #14 Mar 26 2022 15:40:28
%S 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,
%T 14,14,30,3,40,18,32,4,76,9,56,11,40,12,88,15,16,24,32,3,16,34,24,21,
%U 108,6,8,6,24,21,58,12,60,15,16,3,56,30,136,9,16,56
%N a(n) is the length of the cycle of first differences of k such that Fibonacci(k) mod n = k mod n.
%C 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.
%H Lars Blomberg, <a href="/A213277/b213277.txt">Table of n, a(n) for n = 2..23002</a>
%e Example with n=3:
%e 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, ...
%e 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
%e 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
%e Accepted k values indicated by x:
%e x,x,-,-,-,x,-,x,-,-,x,x,x,-,-,-,-,-,-,-,-,-,-,-,x
%e Accepted k values: 0, 1, 5, 7, 10, 11, 12, 14, 24
%e First differences of k values: 1, 4, 2, 3, 1, 1, 2, 10
%e After this the cycle repeats, so a(3) = 8.
%Y Cf. A000045, A213278.
%K nonn
%O 2,1
%A _Lars Blomberg_, Jun 08 2012
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