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 A119283 #29 Jul 29 2024 19:54:20
%S A119283 0,-1,0,-4,5,-20,44,-125,316,-840,2185,-5736,15000,-39289,102840,
%T A119283 -269260,704909,-1845500,4831556,-12649205,33116020,-86698896,
%U A119283 226980625,-594243024,1555748400,-4073002225,10663258224,-27916772500,73087059221,-191344405220,500946156380,-1311494063981,3433536035500,-8989114042584,23533806092185,-61612304234040
%N A119283 Alternating sum of the squares of the first n Fibonacci numbers.
%C A119283 Natural bilateral extension (brackets mark index 0): ..., 840, -316, 125, -44, 20, -5, 4, 0, 1, 0, [0], -1, 0, -4, 5, -20, 44, -125, 316, -840, 2185, ... This is (-A119283)-reversed followed by A119283.
%H A119283 Colin Barker, <a href="/A119283/b119283.txt">Table of n, a(n) for n = 0..1000</a>
%H A119283 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,4,-1,-1)
%F A119283 Let F(n) be the Fibonacci number A000045(n).
%F A119283 a(n) = Sum_{k = 1..n} (-1)^k F(k)^2.
%F A119283 Closed form: a(n) = (-1)^n F(2n+1)/5 - (2 n + 1)/5.
%F A119283 Recurrence: a(n) + a(n-1) - 4 a(n-2) + a(n-3) + a(n-4) = 0.
%F A119283 G.f.: A(x) = (-x - x^2)/(1 + x - 4 x^2 + x^3 + x^4) = -x(1 + x)/((1 - x)^2 (1 + 3 x + x^2)).
%F A119283 a(n) = ( -1-2*n+(-1)^n*( A001906(n+1)-A001906(n) ))/5. - _R. J. Mathar_, Nov 16 2007
%F A119283 a(n) = (2^(-1-n)*(-5*2^(1+n)+5*(-3+sqrt(5))^n - sqrt(5)*(-3+sqrt(5))^n + (-3-sqrt(5))^n*(5+sqrt(5)) - 5*2^(2+n)*n)) / 25. - _Colin Barker_, Apr 25 2017
%e A119283 The first few squares of Fibonacci numbers are: 0, 1, 1, 4, 9, 25, 64, 169.
%e A119283 a(0) = 0.
%e A119283 a(1) = 0 - 1 = -1.
%e A119283 a(2) = 0 - 1 + 1 = 0.
%e A119283 a(3) = 0 - 1 + 1 - 4 = -4.
%e A119283 a(4) = 0 - 1 + 1 - 4 + 9 = 5.
%t A119283 altFiboSqSum[n_Integer] := If[n >= 0, Sum[(-1)^k Fibonacci[k]^2, {k, n}], Sum[-(-1)^k Fibonacci[-k]^2, {k, 1, -n - 1}]]; altFiboSqSum[Range[0, 39]] (* Clary *)
%t A119283 Accumulate[Table[(-1)^n Fibonacci[n]^2, {n, 0, 39}]] (* _Alonso del Arte_, Apr 25 2017 *)
%t A119283 Accumulate[Times@@@Partition[Riffle[Fibonacci[Range[0,40]]^2,{1,-1},{2,-1,2}],2]] (* _Harvey P. Dale_, Jul 29 2024 *)
%o A119283 (PARI) concat(0, Vec(-x*(1 + x) / ((1 - x)^2*(1 + 3*x + x^2)) + O(x^40))) \\ _Colin Barker_, Apr 25 2017
%Y A119283 Cf. A001654, A119282, A119284, A119285, A119286, A119287, A128696, A128698.
%K A119283 sign,easy
%O A119283 0,4
%A A119283 _Stuart Clary_, May 13 2006