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 A156090 #31 Jun 22 2025 00:16:36
%S A156090 0,-4,60,-1096,19640,-352460,6324596,-113490320,2036501104,
%T A156090 -36543529620,655747031980,-11766903046104,211148507797800,
%U A156090 -3788906237314396,67989163763861220,-1220016041512187680,21892299583455516896,-392841376460687116580
%N A156090 Alternating sum of the squares of the first n Fibonacci numbers with index divisible by 3.
%C A156090 Natural bilateral extension (brackets mark index 0): ..., -19640, 1096, -60, 4, 0, [0], -4, 60, -1096, 19640, -352460, ... This is (-a(n))-reversed followed by a(n). That is, a(-n) = -a(n-1).
%H A156090 Muniru A Asiru, <a href="/A156090/b156090.txt">Table of n, a(n) for n = 0..200</a>
%H A156090 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-16,34,-16,-1).
%F A156090 Let F(n) be the n-th Fibonacci number, A000045(n), then: (Start)
%F A156090 a(n) = Sum_{k=1..n} (-1)^k*F(3*k)^2.
%F A156090 Closed form: a(n) = (-1)^n*F(6*n+3)/10 - (2*n + 1)/5.
%F A156090 Recurrence: a(n) + 17*a(n-1) - 17*a(n-2) - a(n-3) = -8.
%F A156090 Recurrence: a(n) + 16*a(n-1) - 34*a(n-2) + 16*a(n-3) + a(n-4) = 0.
%F A156090 G.f.: -4*x*(1 + x)/(1 + 16*x - 34*x^2 + 16*x^3 + x^4) = -4*x(1 + x)/((1 - x)^2*(1 + 18*x + x^2)). (End)
%F A156090 Limit_{n -> oo} a(n)/a(n-1) = -(9 + sqrt(80)). - _A.H.M. Smeets_, Sep 11 2018
%F A156090 E.g.f.: (-1/(5*sqrt(5)))*( exp(-9*x)*(2*sinh(p*x) - sqrt(5)*cosh(p*x)) + sqrt(5)*(1 + 2*x)*exp(x) ), where p = 4*sqrt(5). - _G. C. Greubel_, Jun 12 2025
%p A156090 with(combinat,fibonacci): a:=n->add((-1)^k*fibonacci(3*k)^2,k=1..n): seq(a(n), n=0..20); # _Muniru A Asiru_, Sep 12 2018
%t A156090 a[n_]:= If[n >= 0, Sum[(-1)^k Fibonacci[3 k]^2, {k, 1, n}], Sum[-(-1)^k Fibonacci[-3 k]^2, {k, 1, -n-1}]];
%t A156090 LinearRecurrence[{-16,34,-16,-1}, {0,-4,60,-1096}, 30] (* _Harvey P. Dale_, Oct 24 2016 *)
%o A156090 (GAP) a:=[0,-4,60,-1096];; for n in [5..20] do a[n]:=-16*a[n-1]+34*a[n-2]-16*a[n-3]-a[n-4]; od; a; # _Muniru A Asiru_, Sep 12 2018
%o A156090 (Magma) [(-1)^n*Fibonacci(6*n+3)/10 - (2*n + 1)/5: n in [0..20]]; // _Vincenzo Librandi_, Sep 12 2018
%o A156090 (SageMath)
%o A156090 def A156090(n): return ((-1)^n*fibonacci(6*n+3) -2*(2*n+1))//10 # _G. C. Greubel_, Jun 12 2025
%Y A156090 Cf. A000045, A156084, A156085, A156088, A156089, A156091.
%K A156090 sign,easy
%O A156090 0,2
%A A156090 _Stuart Clary_, Feb 04 2009