|
| |
|
|
A001175
|
|
Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n.
(Formerly M2710 N1087)
|
|
60
| |
|
|
1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, 16, 30, 48, 24, 100, 84, 72, 48, 14, 120, 30, 48, 40, 36, 80, 24, 76, 18, 56, 60, 40, 48, 88, 30, 120, 48, 32, 24, 112, 300, 72, 84, 108, 72, 20, 48, 72, 42, 58, 120, 60, 30, 48, 96, 140, 120, 136
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| These numbers might also have been called Fibonacci periods.
Also, number of perfect multi-Skolem type sequences of order n.
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
|
|
|
REFERENCES
| Michael Baake, Natascha Neumarker and John A. G. Roberts, ORBIT STRUCTURE AND (REVERSING) SYMMETRIES OF TORAL ENDOMORPHISMS ON RATIONAL LATTICES, http://web.maths.unsw.edu.au/~jagr/BNR11.pdf.
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.
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.
Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 459-460.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 162.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly (67 #6, Jun-Jul 1960), pp. 525-532.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
K. S. Brown, Periods of Fibonacci Sequences mod m [broken link]
D. A. Coleman et al., Periods of (q,r)-Fibonacci sequences and Elliptic Curves, Fibonacci Quart. 44, no 1 (2006) 59-70.
G. Nordh, Perfect Skolem sequences
Cyrus Hsia et al., editors Fibonacci residues, Crux Mathematicorum 23:4 pp. 224-226.
N. Patson, Pisano period and permutations of n X n matrices, Australian Math. Soc. Gazette, 2007.
Noel Patson,Square Matrix Permutations [From Noel Patson (n.patson(AT)cqu.edu.au), Mar 28 2010]
M. Renault, Periods of Fibonacci Sequence Modulo m
Eric Weisstein's World of Mathematics, Pisano Number
Wikipedia, Pisano period
|
|
|
FORMULA
| 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
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
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.
a(n) = A001177(n)*A001176(n) for n >= 1. - Henry Bottomley (se16(AT)btinternet.com), Dec 19 2001
|
|
|
MATHEMATICA
| 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)
|
|
|
CROSSREFS
| Cf. A060305 (Fibonacci period mod prime(n)), A003893.
Sequence in context: A098737 A164654 A072396 * A093725 A011413 A010629
Adjacent sequences: A001172 A001173 A001174 * A001176 A001177 A001178
|
|
|
KEYWORD
| nonn,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
