A273624 a(n) = (1/11)*(Fibonacci(4*n) + Fibonacci(6*n)).

1, 15, 248, 4305, 76255, 1361520, 24384737, 437245935, 7843863784, 140737371825, 2525326494911, 45314438127840, 813129752279233, 14590988151618255, 261824431125415640, 4698247224097107345, 84306614992412658847, 1512820749915870503760, 27146466385039244529569

Offset: 1

Author

Peter Bala, May 29 2016

Keywords

• nonn
• easy

Comments

This is a divisibility sequence: if n divides m then a(n) divides a(m). More generally, if r is an even integer then the sequence Fibonacci(r*n) + Fibonacci((r + 2)*n) is a divisibility sequence. See A215466 for the case r = 2.

Also, the sequence s(n) := Fibonacci(4*n) + Fibonacci(6*n) + ... + Fibonacci((2*k + 2)*n) is a divisibility sequence when k is a positive integer that is not a multiple of 3.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..795
• P. Bala, Lucas sequences and divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (25,-128,25,-1).

Crossrefs

Cf. A000045, A049673, A215466, A273623, A273625.

Programs

Magma

[1/11*(Fibonacci(4*n)+Fibonacci(6*n)): n in [1..25]]; // Vincenzo Librandi, Jun 02 2016

Maple

#A273624
with(combinat):
seq(1/11*(fibonacci(4n) + fibonacci(6n)), n = 1..20);

Mathematica

LinearRecurrence[{25,-128,25,-1},{1, 15, 248, 4305},100] (* G. C. Greubel, Jun 02 2016 *)
Table[1/11 (Fibonacci[4 n] + Fibonacci[6 n]), {n, 1, 30}] (* Vincenzo Librandi, Jun 02 2016 *)

PARI

a(n)=(fibonacci(4*n) + fibonacci(6*n))/11 \\ Charles R Greathouse IV, Jun 08 2016

Formula

a(n) = -a(-n).

a(n) = 25*a(n-1) - 128*a(n-2) + 25*a(n-3) - a(n-4).

O.g.f. (x^2 - 10*x + 1)/((x^2 - 7*x + 1)*(x^2 - 18*x + 1)).