A348591 a(n) = L(n)*L(n+1) mod F(n+2) where F=A000045 is the Fibonacci numbers and L = A000032 is the Lucas numbers.

0, 1, 0, 3, 5, 3, 18, 3, 52, 3, 141, 3, 374, 3, 984, 3, 2581, 3, 6762, 3, 17708, 3, 46365, 3, 121390, 3, 317808, 3, 832037, 3, 2178306, 3, 5702884, 3, 14930349, 3, 39088166, 3, 102334152, 3, 267914293, 3, 701408730, 3, 1836311900, 3, 4807526973, 3, 12586269022, 3, 32951280096, 3, 86267571269, 3

Offset

0,4

Links

Robert Israel, Table of n, a(n) for n = 0..4761

Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-1,-1).

Formula

a(n) = 3 if n >= 3 is odd.

a(n) = A000045(n+2)-3 if n >= 2 is even.

a(n) + a(n+1) - 3*a(n+2) - 3*a(n+3) + a(n+4) + a(n+5) = 0 for n >= 2.

G.f.: -x*(2*x^5-5*x^3-x-1)/((x+1)*(x^2+x-1)*(x^2-x-1)). - Alois P. Heinz, Jan 26 2022

Example

a(5) = L(5)*L(6) mod F(7) = 11*18 mod 13 = 3.

Maple

F:= combinat:-fibonacci:
L:= n -> F(n-1)+F(n+1):
map(n -> L(n)*L(n+1) mod F(n+2), [$0..30]);

Mathematica

a[n_] := Mod[LucasL[n] * LucasL[n + 1], Fibonacci[n + 2]]; Array[a, 50, 0] (* Amiram Eldar, Jan 26 2022 *)

Prog

(Python)
from gmpy2 import fib, lucas2
def A348591(n): return (lambda x, y:int(x[0]*x[1] % y))(lucas2(n+1), fib(n+2)) # Chai Wah Wu, Jan 26 2022

Crossrefs

Cf. A000032, A000045, A215602, A333599, A347861, A348592.

Sequence in context: A258811 A066845 A014782 * A094466 A134771 A080349

Adjacent sequences: A348588 A348589 A348590 * A348592 A348593 A348594

Keyword

nonn,easy

Author

J. M. Bergot and Robert Israel, Jan 25 2022

Status

approved