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A253262 Expansion of (x + x^2 + x^3) / (1 - x + x^2 - x^3 + x^4) in powers of x.

%I A253262 #22 Sep 08 2022 08:46:10
%S A253262 0,1,2,2,1,0,-1,-2,-2,-1,0,1,2,2,1,0,-1,-2,-2,-1,0,1,2,2,1,0,-1,-2,-2,
%T A253262 -1,0,1,2,2,1,0,-1,-2,-2,-1,0,1,2,2,1,0,-1,-2,-2,-1,0,1,2,2,1,0,-1,-2,
%U A253262 -2,-1,0,1,2,2,1,0,-1,-2,-2,-1,0,1,2,2,1,0,-1
%N A253262 Expansion of (x + x^2 + x^3) / (1 - x + x^2 - x^3 + x^4) in powers of x.
%C A253262 Cycle period is 10. - _Robert G. Wilson v_, Aug 02 2018
%H A253262 G. C. Greubel, <a href="/A253262/b253262.txt">Table of n, a(n) for n = 0..2500</a>
%H A253262 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1).
%F A253262 Euler transform of length 10 sequence [2, -1, -1, 0, -1, 0, 0, 0, 0, 1].
%F A253262 G.f.: x * (1 + x) * (1 - x^3) / ((1 - x) * (1 + x^5)).
%F A253262 INVERT transform is A003476.
%F A253262 a(n) = -a(-n) = -a(n+5) for all n in Z.
%F A253262 a(n) = f(n) / f(1) where f(n) := tan( am( n*x, m)) where x = 0.7379409146... and m = 1.3481185591... and am() is the Jacobi amplitude function.
%e A253262 G.f. = x + 2*x^2 + 2*x^3 + x^4 - x^6 - 2*x^7 - 2*x^8 - x^9 + x^11 + 2*x^12 + ...
%t A253262 a[ n_] := {1, 2, 2, 1, 0}[[Mod[n, 5, 1]]] (-1)^Quotient[n, 5];
%t A253262 CoefficientList[Series[x*(1+x)*(1-x^3)/((1-x)*(1+x^5)), {x,0,60}], x] (* _G. C. Greubel_, Aug 02 2018 *)
%t A253262 CoefficientList[ Series[x (x^2 + x + 1)/(x^4 - x^3 + x^2 - x + 1), {x, 0, 75}], x] (* or *)
%t A253262 LinearRecurrence[{1, -1, 1, -1}, {0, 1, 2, 2}, 75] (* _Robert G. Wilson v_, Aug 02 2018 *)
%o A253262 (PARI) {a(n) = [0, 1, 2, 2, 1][n%5 + 1] * (-1)^(n\5)};
%o A253262 (PARI) x='x+O('x^60); concat([0], Vec(x*(1+x)*(1-x^3)/((1-x)*(1+x^5)))) \\ _G. C. Greubel_, Aug 02 2018
%o A253262 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1-x^3)/((1-x)*(1+x^5)))); // _G. C. Greubel_, Aug 02 2018
%Y A253262 Cf. A003476.
%K A253262 sign,easy
%O A253262 0,3
%A A253262 _Michael Somos_, Apr 30 2015