A214884 a(n) = Sum_{k=0..n} (-1)^k*F(k)*F(k+2), where F=A000045 (Fibonacci numbers).

0, -2, 1, -9, 15, -50, 118, -324, 831, -2195, 5725, -15012, 39276, -102854, 269245, -704925, 1845483, -4831574, 12649186, -33116040, 86698875, -226980647, 594243001, -1555748424, 4073002200, -10663258250, 27916772473, -73087059249, 191344405191

Offset: 0

Author

Wolfdieter Lang, Jul 30 2012

Keywords

• sign
• easy

Comments

The present sequence is the m=2 member of the m-family of sequences b(m,n):=Sum_{k=0..n} (-1)^k*F(k+2)*F(k) given by b(m,n) = (L(m)*A119283(n) + F(m)*(-1)^n*A001654(n))/2, with A119283(n) = b(0,n) = ((-1)^n*F(2*n+1) - (2*n+1))/5 and A001654(n) = F(n+1)*F(n), where F and L are the Fibonacci and Lucas numbers, A000045 and A000032, respectively.

The o.g.f. of b(m,n) is A(m,x) = -(1/2)*x*(F(m+1) + F(m-1)*x)/((1-x)^2*(1+3*x+x^2)), m >= 0, with F(-1) = 1. For the unsigned sums see a comment on A080144.

b(m, n) = ((-1)^n*F(m + 2*n + 1) - n*L(m) - F(m + 1))/5. - Ehren Metcalfe, Aug 21 2017

Crossrefs

Cf. A119283, -A077916(n-1) for the m=0 and m=1 cases. A214885 for m=3.

Programs

Mathematica

Table[Sum[(-1)^k*Fibonacci[k]*Fibonacci[k + 2], {k, 0, n}], {n, 0, 28}] (* Michael De Vlieger, Aug 23 2017 *)

Formula

a(n) = b(2,n) = (3*A119283(n) + (-1)^n*A001654(n))/2, n >= 0.

O.g.f.: -x*(2+x)/((1-x)^2*(1+3*x+x^2)) (see the comment section).

a(n) = ((-1)^n*Fibonacci(2*n + 3) - 3*n - 2)/5. - Ehren Metcalfe, Aug 21 2017