A236395 -id:A236395 - cp's OEIS frontend

A132634 a(n) = Fibonacci(n) mod n^2.

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 0, 64, 181, 160, 219, 152, 316, 210, 365, 362, 287, 91, 288, 25, 389, 317, 291, 378, 440, 869, 261, 574, 339, 765, 432, 443, 533, 1285, 1355, 1641, 1504, 85, 1741, 20, 551, 1832, 576, 1457, 1525, 389, 803, 2066, 332, 1820, 245

Offset: 1

Hieronymus Fischer, Aug 24 2007

nonn

a(n)=0 for n=1 and n=12 only (conjecture).

			a(13) = 64, since Fibonacci(13) = 233 == 64 (mod 13^2).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from Hieronymus Fischer)

Cf. A000045, A023173, A236395.

p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, n, n^2)[1, 2]:
seq(a(n), n=1..80);

Table[Mod[Fibonacci[n],n^2],{n,200}] (* Vladimir Joseph Stephan Orlovsky, Nov 28 2010 *)

A051831 a(n) = Fibonacci(prime(n)) mod prime(n), where prime(n) is the n-th prime.

Original entry on oeis.org

1, 2, 0, 6, 1, 12, 16, 1, 22, 1, 1, 36, 1, 42, 46, 52, 1, 1, 66, 1, 72, 1, 82, 1, 96, 1, 102, 106, 1, 112, 126, 1, 136, 1, 1, 1, 156, 162, 166, 172, 1, 1, 1, 192, 196, 1, 1, 222, 226, 1, 232, 1, 1, 1, 256, 262, 1, 1, 276, 1, 282, 292, 306, 1, 312, 316, 1, 336, 346, 1, 352, 1

Offset: 1

Jud McCranie, Dec 11 1999

nonn

Terms are 1 when prime(n) == 1 or 4 mod 5, terms are prime(n)-1 when prime(n) == 2 or 3 mod 5.

In general, it appears that Fibonacci(k*p) mod p = Fibonacci(k) or p-Fibonacci(k) for prime p > Fibonacci(k). For example Fibonacci(8*29) mod 29 = 21. - Gary Detlefs, May 28 2014

			prime(3) = 5, fibonacci(5) = 5 == 0 mod 5.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
R. Peele and P. Stanica, Matrix powers of column-justified Pascal triangles and Fibonacci sequences, arXiv:math/0010186 [math.CO], 2000.

Cf. A000045, A003631, A045468, A051834, A051830, A236395.

p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, ithprime(n)$2)[1, 2]:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 10 2015

Mod[Fibonacci[Prime[#]],Prime[#]]&/@Range[75] (* Harvey P. Dale, Jan 14 2011 *)

vector(80, n, fibonacci(prime(n)) % prime(n)) \\ Michel Marcus, Jul 15 2015

cp's OEIS Frontend

A132634 a(n) = Fibonacci(n) mod n^2.

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