A292612 a(n) = F(n)^2 + 4*(-1)^n = F(n+3)*F(n-3), where F = A000045.

4, -3, 5, 0, 13, 21, 68, 165, 445, 1152, 3029, 7917, 20740, 54285, 142133, 372096, 974173, 2550405, 6677060, 17480757, 45765229, 119814912, 313679525, 821223645, 2149991428, 5628750621, 14736260453, 38580030720, 101003831725, 264431464437, 692290561604, 1812440220357

Offset: 0

Author

Bruno Berselli, Sep 20 2017

Keywords

• sign
• easy

Comments

This is the case k=3 of the identity F(n)^2 - F(k)^2*(-1)^(n+k) = F(n+k)*F(n-k), known also as Catalan's identity.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Catalan's Identity.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A000045, A001622, A005248: Lucas(2*n), A001654: F(n)*F(n+1).

Cf. A007598 (k=0), A059929 (k=1, without initial 1), A192883 (k=2, without initial -1), this sequence (k=3).

Programs

GAP

List([0..10^2],n ->Fibonacci(n)^2+4*(-1)^n); # Muniru A Asiru, Sep 26 2017

Magma

[Fibonacci(n)^2+4*(-1)^n: n in [0..40]];

Maple

with(combinat,fibonacci):  A292612:=seq(fibonacci(n)^2+4*(-1)^n, n=0..10^2); # Muniru A Asiru, Sep 26 2017

Mathematica

Table[Fibonacci[n]^2 + 4 (-1)^n, {n, 0, 40}]

PARI

for(n=0, 40, print1(fibonacci(n)^2+4*(-1)^n", "));

PARI

Vec((4-11*x+3*x^2)/((1+x)*(1-3*x+x^2))+O(x^30)) \\ Colin Barker, Sep 20 2017

Sage

[fibonacci(n)^2+4*(-1)^n for n in range(40)]

Formula

G.f.: (4 - 11*x + 3*x^2)/((1 + x)*(1 - 3*x + x^2)).

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

a(n) = 4*A001654(n+1) - 11*A001654(n) + 3*A001654(n-1) with A001654(-1)=0.

5*a(n) = Lucas(2*n) + 18*(-1)^n. Note that Lucas(2*n) + r*(-1)^n is divisible by 5 for r = -2, 3, -7, 8, -12, 13, -17, 18, -22, 23, -27, ... = (-1/4)*(3 + 5*(2*m+1)*(-1)^m) = (-1)^m*A047221(m). On the other hand, a(n) is divisible by 5 when n is a member of A047221.

a(n) = (1/5)*(18*(-1)^n + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n). - Colin Barker, Sep 20 2017

Sum_{n>=4} 1/a(n) = 143/960. - Amiram Eldar, Oct 05 2020

Sum_{n>=4} (-1)^n/a(n) = 3/(4*phi) - 407/960, where phi is the golden ratio (A001622). - Amiram Eldar, Oct 06 2020