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 A317499 #20 Aug 08 2025 07:05:22
%S A317499 1,-2,4,-5,4,4,-23,58,-104,139,-104,-104,625,-1562,2812,-3749,2812,
%T A317499 2812,-16871,42178,-75920,101227,-75920,-75920,455521,-1138802,
%U A317499 2049844,-2733125,2049844,2049844,-12299063,30747658,-55345784,73794379,-55345784,-55345784
%N A317499 Coefficients in expansion of 1/(1 + 2*x - 3*x^3).
%C A317499 The coefficients in the expansion of 1/(1 + 2*x - 3*x^3) are given by the sequence generated by the row sums in triangle A317503.
%C A317499 Coefficients in expansion of 1/(1 + 2*x - 3*x^3) are given by the sum of numbers along second Layer skew diagonals pointing top-left in triangle A303901 ((3-2*x)^n) and by the sum of numbers along second Layer skew diagonals pointing top-right in triangle A317498 ((-2+3*x)^n), see links.
%D A317499 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 396, 397.
%H A317499 Colin Barker, <a href="/A317499/b317499.txt">Table of n, a(n) for n = 0..1000</a>
%H A317499 Zagros Lalo, <a href="/A317499/a317499.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 - 2x)^n</a>
%H A317499 Zagros Lalo, <a href="/A317499/a317499_1.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3x)^n</a>
%H A317499 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,3).
%F A317499 a(0)=1, a(n) = -2*a(n-1) + 3*a(n-3) for n = 0,1...; a(n)=0 for n < 0.
%F A317499 a(n) = (2^(-n)*(2^n + (-3-i*sqrt(3))^n*(3-2*i*sqrt(3)) + (-3+i*sqrt(3))^n*(3+2*i*sqrt(3)))) / 7 where i=sqrt(-1). - _Colin Barker_, Aug 02 2018
%p A317499 seq(coeff(series(1/(1+2*x-3*x^3), x,n+1),x,n),n=0..40); # _Muniru A Asiru_, Aug 01 2018
%t A317499 CoefficientList[Series[1/(1 + 2 x - 3 x^3), {x, 0, 40}], x]
%t A317499 a[0] = 1; a[n_] := a[n] = If[n < 0, 0, -2 * a[n - 1] + 3 * a[n - 3]]; Table[a[n], {n, 0, 40}] // Flatten
%t A317499 LinearRecurrence[{-2, 0, 3}, {1, -2, 4}, 41]
%o A317499 (GAP) a:=[1,-2,4];; for n in [4..40] do a[n]:=-2*a[n-1]+3*a[n-3]; od; a; # _Muniru A Asiru_, Aug 01 2018
%o A317499 (PARI) Vec(1 / ((1 - x)*(1 + 3*x + 3*x^2)) + O(x^40)) \\ _Colin Barker_, Aug 02 2018
%Y A317499 Cf. A317502, A317503.
%Y A317499 Cf. A303901, A317498.
%K A317499 sign,easy
%O A317499 0,2
%A A317499 _Zagros Lalo_, Jul 31 2018