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A257543 Expansion of 1 / (1 - x^5 - x^8 + x^9) in powers of x.

%I A257543 #16 Sep 08 2022 08:46:12
%S A257543 1,0,0,0,0,1,0,0,1,-1,1,0,0,2,-2,1,1,-2,4,-3,1,3,-6,7,-3,-2,9,-13,11,
%T A257543 -1,-11,22,-23,12,10,-33,46,-35,2,43,-78,81,-37,-41,122,-159,118,4,
%U A257543 -162,281,-277,114,167,-443,558,-391,-52,610,-1001,949,-338,-662
%N A257543 Expansion of 1 / (1 - x^5 - x^8 + x^9) in powers of x.
%H A257543 G. C. Greubel, <a href="/A257543/b257543.txt">Table of n, a(n) for n = 0..2500</a>
%H A257543 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1,0,0,1,-1).
%F A257543 G.f.: 1 / ((1 - x^4) * (1 + x^4 - x^5)) = (1 + x) / ((1 + x^3) * (1 - x^4) * (1 + x - x^3)).
%F A257543 a(n) = a(n-5) + a(n-8) - a(n-9) for all n in Z.
%F A257543 a(n) - a(n+2) - a(n+3) has period 12.
%F A257543 a(n) - a(n+12) = A104769(n+5) = -A247917(n+4) for all n in Z.
%F A257543 a(n) + a(n+1) = A247918(n) for all n in Z.
%F A257543 a(n) = -A233522(-9 - n) for all n in Z.
%e A257543 G.f. = 1 + x^5 + x^8 - x^9 + x^10 + 2*x^13 - 2*x^14 + x^15 + x^16 + ...
%t A257543 a[ n_] := SeriesCoefficient[ If[ n >= 0, 1 / (1 - x^5 - x^8 + x^9), -x^9 /(1 - x - x^4 + x^9)], {x, 0, Abs@n}];
%o A257543 (PARI) {a(n) = if( n>=0, polcoeff( 1 / (1 - x^5 - x^8 + x^9) + x * O(x^n), n), polcoeff( -x^9 / (1 - x - x^4 + x^9) + x * O(x^-n), -n))};
%o A257543 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 / ((1-x^4)*(1+x^4-x^5)))); // _G. C. Greubel_, Aug 02 2018
%Y A257543 Cf. A104769, A233522, A247917, A247918.
%K A257543 sign,easy
%O A257543 0,14
%A A257543 _Michael Somos_, Apr 28 2015