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A015135 Consider Fibonacci-type sequences f(0)=X, f(1)=Y, f(k)=f(k-1)+f(k-2) mod n; all are periodic; sequence gives number of distinct periods. 1
1, 2, 2, 3, 3, 4, 2, 4, 3, 6, 3, 5, 2, 4, 5, 5, 2, 4, 3, 7, 3, 6, 2, 6, 4, 4, 4, 5, 3, 10, 3, 6, 5, 3, 5, 5, 2, 4, 4, 7, 2, 6, 2, 7, 7, 3, 2, 6, 3, 8, 4, 5, 2, 5, 5, 6, 5, 6, 3, 11, 2, 4, 5, 7, 5, 10, 2, 4, 3, 10, 3, 6, 2, 4, 7, 5, 5, 8, 3, 9, 5, 4, 2, 7, 5, 4, 5, 9, 2, 10, 4, 4, 5, 4, 7, 7, 2, 6, 7, 9, 3, 6, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Consider the 2-step recursion f(k)=f(k-1)+f(k-2) mod n. For any of the n^2 initial conditions f(1) and f(2) in Zn, the recursion has a finite period. Each of these n^2 vectors belongs to exactly one orbit. In general, there are only a few different orbit lengths for each n. For n=8, there are 4 different lengths: 1, 3, 6 and 12. The maximum possible length of an orbit is A001175(n), the period of the Fibonacci 2-step sequence mod n. - T. D. Noe, May 02 2005
LINKS
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
CROSSREFS
Cf. A015134 (orbits of 2-step sequences), A106306 (primes that yield a simple orbit structure in 2-step recursions).
Sequence in context: A106494 A356555 A339811 * A116619 A337776 A366611
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jan 06 2005
STATUS
approved

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Last modified June 17 15:57 EDT 2024. Contains 373463 sequences. (Running on oeis4.)