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 A014292 #22 Sep 08 2022 08:44:39
%S A014292 0,0,1,2,3,4,4,2,-3,-12,-25,-40,-52,-52,-27,38,155,324,520,678,681,
%T A014292 360,-481,-2000,-4200,-6760,-8839,-8918,-4797,6084,25804,54442,87877,
%U A014292 115228,116775,63880,-76892,-332892,-705667,-1142322
%N A014292 a(n) = 2*a(n-1) - a(n-2) - a(n-4) with a(0) = a(1) = 0, a(2) = 1, a(3) = 2.
%C A014292 Imaginary part of the sequence of complex numbers defined by c(0) = 1, c(1) = 1, for n>1 c(n) = c(n-1) + i*c(n-2). - _Gerald McGarvey_, Apr 24 2005
%C A014292 a(n) = sqrt(3)*y where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)*(0,s,s,s) where s = 1/sqrt(3). - _Gerald McGarvey_, Apr 25 2005
%C A014292 For n>=1, a(n) is equal to -1 times the imaginary part of the determinant of the n X n matrix with the sqrt(i)'s along the superdiagonal and the subdiagonal (i is the imaginary unit), 1's along the main diagonal, and 0's everywhere else (see Mathematica code below). - _John M. Campbell_, Jun 04 2011
%H A014292 G. C. Greubel, <a href="/A014292/b014292.txt">Table of n, a(n) for n = 0..1000</a>
%H A014292 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,-1).
%F A014292 a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n-k+2, k)*sin(Pi*k/2). - _Paul Barry_, Apr 25 2005
%F A014292 G.f.: x^2/(1 - 2*x + x^2 + x^4). - _R. J. Mathar_, Oct 22 2008
%t A014292 Table[-Im[Det[Array[KroneckerDelta[#1 + 1, #2]*Sqrt[I] &, {n, n}] + Array[KroneckerDelta[#1 - 1, #2]*Sqrt[I] &, {n, n}] + IdentityMatrix[n]]], {n, 1, 40}] (* _John M. Campbell_, Jun 04 2011 *)
%t A014292 LinearRecurrence[{2,-1,0,-1}, {0,0,1,2}, 40] (* _G. C. Greubel_, Jun 12 2019 *)
%o A014292 (PARI) my(x='x+O('x^40)); concat([0,0], Vec(x^2/(1-2*x+x^2+x^4))) \\ _G. C. Greubel_, Jun 12 2019
%o A014292 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2/(1-2*x+x^2+x^4) )); // _G. C. Greubel_, Jun 12 2019
%o A014292 (Sage) (x^2/(1-2*x+x^2+x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 12 2019
%o A014292 (GAP) a:=[0,0,1,2];; for n in [5..40] do a[n]:=2*a[n-1]-a[n-2]-a[n-4]; od; a; # _G. C. Greubel_, Jun 12 2019
%Y A014292 Cf. A104862, A104862.
%K A014292 sign
%O A014292 0,4
%A A014292 _N. J. A. Sloane_