This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A214884 #15 Aug 26 2017 17:54:20
%S A214884 0,-2,1,-9,15,-50,118,-324,831,-2195,5725,-15012,39276,-102854,269245,
%T A214884 -704925,1845483,-4831574,12649186,-33116040,86698875,-226980647,
%U A214884 594243001,-1555748424,4073002200,-10663258250,27916772473,-73087059249,191344405191
%N A214884 a(n) = Sum_{k=0..n} (-1)^k*F(k)*F(k+2), where F=A000045 (Fibonacci numbers).
%C A214884 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.
%C A214884 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.
%C A214884 b(m, n) = ((-1)^n*F(m + 2*n + 1) - n*L(m) - F(m + 1))/5. - _Ehren Metcalfe_, Aug 21 2017
%F A214884 a(n) = b(2,n) = (3*A119283(n) + (-1)^n*A001654(n))/2, n >= 0.
%F A214884 O.g.f.: -x*(2+x)/((1-x)^2*(1+3*x+x^2)) (see the comment section).
%F A214884 a(n) = ((-1)^n*Fibonacci(2*n + 3) - 3*n - 2)/5. - _Ehren Metcalfe_, Aug 21 2017
%t A214884 Table[Sum[(-1)^k*Fibonacci[k]*Fibonacci[k + 2], {k, 0, n}], {n, 0, 28}] (* _Michael De Vlieger_, Aug 23 2017 *)
%Y A214884 Cf. A119283, -A077916(n-1) for the m=0 and m=1 cases. A214885 for m=3.
%K A214884 sign,easy
%O A214884 0,2
%A A214884 _Wolfdieter Lang_, Jul 30 2012