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 A091005 #27 Dec 14 2024 03:59:30
%S A091005 0,0,1,-1,7,-13,55,-133,463,-1261,4039,-11605,35839,-105469,320503,
%T A091005 -953317,2876335,-8596237,25854247,-77431669,232557151,-697147165,
%U A091005 2092490071,-6275373061,18830313487,-56482551853,169464432775,-508359743893,1525146340543
%N A091005 Expansion of x^2/((1-2*x)*(1+3*x)).
%C A091005 Inverse binomial transform of A091002.
%H A091005 G. C. Greubel, <a href="/A091005/b091005.txt">Table of n, a(n) for n = 0..1000</a>
%H A091005 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,6).
%F A091005 2^n = A091003(n) + 3*A091004(n) + 6*a(n).
%F A091005 a(n) = (3*2^n + 2*(-3)^n - 5*0^n)/30.
%F A091005 E.g.f.: (3*exp(2*x) + 2*exp(-3*x) - 5)/30. - _G. C. Greubel_, Feb 01 2019
%t A091005 a[n_]:=(MatrixPower[{{1,4},{1,-2}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)
%t A091005 Join[{0, 0}, LinearRecurrence[{-1, 6}, {1, -1}, 30]] (* _G. C. Greubel_, Feb 01 2019 *)
%t A091005 CoefficientList[Series[x^2/((1-2x)(1+3x)),{x,0,30}],x] (* _Harvey P. Dale_, Apr 30 2022 *)
%o A091005 (PARI) vector(30, n, n--; (3*2^n + 2*(-3)^n - 5*0^n)/30) \\ _G. C. Greubel_, Feb 01 2019
%o A091005 (Magma) [0] cat [(3*2^n + 2*(-3)^n)/30: n in [1..30]]; // _G. C. Greubel_, Feb 01 2019
%o A091005 (Sage) [0] + [(3*2^n + 2*(-3)^n)/30 for n in (1..30)] # _G. C. Greubel_, Feb 01 2019
%o A091005 (GAP) Concatenation([0], List([1..30], n -> (3*2^n + 2*(-3)^n)/30)); # _G. C. Greubel_, Feb 01 2019
%Y A091005 Cf. A015441.
%K A091005 easy,sign
%O A091005 0,5
%A A091005 _Paul Barry_, Dec 13 2003