%I A001175 M2710 N1087
%S A001175 1,3,8,6,20,24,16,12,24,60,10,24,28,48,40,24,36,24,18,60,16,30,48,24,
%T A001175 100,84,72,48,14,120,30,48,40,36,80,24,76,18,56,60,40,48,88,30,120,48,
%U A001175 32,24,112,300,72,84,108,72,20,48,72,42,58,120,60,30,48,96,140,120,136
%N A001175 Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n.
%C A001175 Also, number of perfect multi-Skolem type sequences of order n.
%C A001175 Index the Fibonacci numbers so that 3 is the fourth number. If the modulo 
               base is a Fibonacci number (>=3) with an even index, the period is 
               twice the index. If the base is a Fibonacci number (>=5) with an 
               odd index, the period is 4 times the index. - Kerry Mitchell (lkmitch(AT)gmail.com), 
               Dec 11 2005
%D A001175 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001175 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001175 Crux Mathematicorum, Fibonacci Residues, 1997 Vol. 23 No. 4 pp. 224-6 
               CMS
%D A001175 J. D. Fulton and W. L. Morris, On arithmetical functions related to the 
               Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.
%D A001175 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.
%D A001175 Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 
               459-460.
%D A001175 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 162.
%D A001175 D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly (67 #6, Jun-Jul 
               1960), pp. 525-532.
%H A001175 T. D. Noe, <a href="b001175.txt">Table of n, a(n) for n = 1..10000</a>
%H A001175 K. S. Brown, <a href="http://www.mathpages.com/home/kmath078.htm">Periods 
               of Fibonacci Sequences mod m</a>
%H A001175 D. A. Coleman et al., <a href="http://ww2.lafayette.edu/~reiterc/nt/qr_fib_ec_preprint.pdf">
               Periods of (q,r)-Fibonacci sequences and Elliptic Curves</a>, Fibonacci 
               Quart. 44, no 1 (2006) 59-70.
%H A001175 G. Nordh, <a href="http://arXiv.org/abs/math.NT/0506155">Perfect Skolem 
               sequences</a>
%H A001175 N. Patson, <a href="http://www.austms.org.au/Gazette/2007/Mar07/39PatsonPisano.pdf">
               Pisano period and permutations of n X n matrices</a>, Australian 
               Math. Soc. Gazette, 2007.
%H A001175 M. Renault, <a href="http://www.math.temple.edu/~renault/fibonacci/fiblist.html">
               Periods of Fibonacci Sequence Modulo m</a>
%H A001175 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PisanoPeriod.html">Pisano Number</a>
%F A001175 Let the prime factorization of n be p1^e1...pk^ek. Then a(n) = lcm(a(p1^e1), 
               ..., a(pk^ek)). - T. D. Noe (noe(AT)sspectra.com), May 02 2005
%F A001175 a(n) = n-1 if n is a prime > 5 included in A003147 ( n = 11, 19, 31, 
               41, 59, 61, 71, 79, 109...) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 04 2002
%F A001175 K. S. Brown shows that a(n)/n <= 6 for all n and a(n)=6n if and only 
               if n has the form 2*5^k.
%F A001175 a(n) = A001177(n)*A001176(n) for n >= 1. - Henry Bottomley (se16(AT)btinternet.com), 
               Dec 19 2001
%t A001175 Table[a={1, 0}; a0=a; k=0; While[k++; s=Mod[Plus@@a, n]; a=RotateLeft[a]; 
               a[[2]]=s; a!=a0]; k, {n, 2, 100}] (Noe)
%Y A001175 Cf. A060305 (Fibonacci period mod prime(n)).
%Y A001175 Sequence in context: A098737 A164654 A072396 this_sequence A093725 A011413 
               A010629
%Y A001175 Adjacent sequences: A001172 A001173 A001174 this_sequence A001176 A001177 
               A001178
%K A001175 nonn,nice
%O A001175 1,2
%A A001175 N. J. A. Sloane (njas(AT)research.att.com).