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A001178 Fibonacci frequency of n.

%I A001178 M3207 N1298 #37 Feb 28 2022 14:59:57
%S A001178 0,4,3,2,3,1,2,2,1,2,3,1,3,2,3,1,2,1,2,2,2,2,2,0,3,3,2,2,3,1,2,2,3,2,
%T A001178 2,1,3,2,3,2,3,2,3,2,1,2,3,1,3,2,2,3,3,2,3,2,2,3,4,1,2,2,2,3,3,1,3,2,2
%N A001178 Fibonacci frequency of n.
%C A001178 a(A235702(n)) = 0. - _Reinhard Zumkeller_, Jan 15 2014
%C A001178 a(n) is the least nonnegative integer k such that the function iterates f: {1, 2, ...} -> {1, 2, ...}, n -> f(n) = A001175(n), satisfy f^[k+1](n) = f^[k](n), where f^[0] is the identity map f^[0](n) = n and f^[k+1] = f o f^[k]. See the Fulton and Morris link, where the function f is called pi and a(n)= omega(n) for n >= 2, and omega(24) should be 0. (see the Zumkeller remark on the Hannon and Morris reference) - _Wolfdieter Lang_, Jan 18 2015
%D A001178 B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. Report ORNL-4261, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Jun 1968. [There is a typo in the value of a(24) given in the table on the last page.]
%D A001178 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001178 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001178 Reinhard Zumkeller, <a href="/A001178/b001178.txt">Table of n, a(n) for n = 1..10000</a>
%H A001178 D. Fulton and W. L. Morris, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf">On arithmetical functions related to the Fibonacci numbers</a>, Acta Arithmetica, 16 (1969), 105-110.
%H A001178 B. H. Hannon and W. L. Morris, <a href="/A001175/a001175.pdf">Tables of Arithmetical Functions Related to the Fibonacci Numbers</a> [Annotated and scanned copy]
%H A001178 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pisano_period">Pisano period</a>
%F A001178 See a comment above and the program.
%t A001178 pi[1] = 1;
%t A001178 pi[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[ Fibonacci[k+1], n] == 1, Return[k]]];
%t A001178 a[n_] := Length[FixedPointList[pi, n]] - 2;
%t A001178 a /@ Range[100] (* _Jean-François Alcover_, Oct 28 2019 *)
%o A001178 (Haskell)
%o A001178 a001178 = f 0 where
%o A001178    f j x = if x == y then j else f (j + 1) y  where y = a001175 x
%o A001178 -- _Reinhard Zumkeller_, Jan 15 2014
%o A001178 (Python)
%o A001178 from itertools import count
%o A001178 def A001178(n): # uses function from A001175
%o A001178     m = n
%o A001178     for c in count(0):
%o A001178         k = A001175(m)
%o A001178         if k == m:
%o A001178             return c
%o A001178         m = k # _Chai Wah Wu_, Feb 28 2022
%Y A001178 Cf. A001175, A235702.
%K A001178 nonn
%O A001178 1,2
%A A001178 _N. J. A. Sloane_
%E A001178 a(24) corrected by _Reinhard Zumkeller_, Jan 15 2014