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A015134
Consider Fibonacci-type sequences b(0)=X, b(1)=Y, b(k)=b(k-1)+b(k-2) mod n; all are periodic; sequence gives number of distinct periods.
13
1, 2, 2, 4, 3, 4, 4, 8, 5, 6, 14, 10, 7, 8, 12, 16, 9, 16, 22, 16, 29, 28, 12, 30, 13, 14, 14, 22, 63, 24, 34, 32, 39, 34, 30, 58, 19, 86, 32, 52, 43, 58, 22, 78, 39, 46, 70, 102, 25, 26, 42, 40, 27, 52, 160, 74, 63, 126, 62, 70, 63, 134, 104, 64, 57, 78, 34, 132, 101, 60, 74, 222
OFFSET
1,2
COMMENTS
b(k) >= k/4 (by counting zeros). - R. C. Johnson (bob.johnson(AT)dur.ac.uk), Nov 20 2003
LINKS
Peter McAndrew, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David Radcliffe)
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.
FORMULA
a(2^n) = 2^n. - Thomas Anton, Apr 16 2023
Conjectures from Stephen Bartell, Aug 20 2023: (Start)
a(3 * 2^n) = 3*(3 * 2^n - 14), n >= 3.
a(3^n) = (3^n + 1)/2, n >= 0.
a(2 * 3^n) = 2*(3^n - 1), n >= 1.
a(5 * 3^n) = (3^(n+2) - 3)/2, n >= 0.
a(10 * 3^n) = 6(3^(n+1) - 5), n >= 1.
a(5^n) = (5^n + 1)/2, n >= 0.
a(2 * 5^n) = 5^n + 1, n >= 0.
a(3 * 5^n) = (5^(n+1) - 1)/2, n >= 0.
a(4 * 5^n) = 3 * 5^n + 1, n >= 0.
a(6 * 5^n) = 5^(n+1) - 1, n >= 0.
a(7^n) = (7^n + 1)/2, n >= 0.
a(2 * 7^n) = 7^n + 1, n >= 0. (End)
Conjectures from Stephen Bartell, Aug 16 2024: (Start)
a(11^n) = (6*11^n - 1)/5 + n, n >= 0.
a(13^n) = (13^n + 1)/2, n >= 0.
a(17^n) = (17^n + 1)/2, n >= 0.
a(19^n) = (10*19^n - 1)/9 + n, n >= 0.
a(23^n) = (23^n + 1)/2, n >= 0.
a(29^n) = (15*29^n - 8)/7 + 2n, n >= 0.
For prime p not in A053032, a(p^n) = 1 + ((p+1)(p^n-1))/(A001175(p)) [except for p = 5].
For prime p in A053032, a(p^n) = 1 + ((p+1)(p^n-1)+n(p-1))/(A001175(p)). (End)
CROSSREFS
Cf. A015135 (number of different orbit lengths of the 2-step recursion mod n), A106306 (primes that yield a simple orbit structure in 2-step recursions).
Sequence in context: A046701 A140472 A109168 * A171580 A246796 A177235
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jan 06 2005
STATUS
approved