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A247917 Expansion of 1 / (1 + x - x^3) in powers of x.

%I A247917 #20 Apr 10 2025 17:24:23
%S A247917 1,-1,1,0,-1,2,-2,1,1,-3,4,-3,0,4,-7,7,-3,-4,11,-14,10,1,-15,25,-24,9,
%T A247917 16,-40,49,-33,-7,56,-89,82,-26,-63,145,-171,108,37,-208,316,-279,71,
%U A247917 245,-524,595,-350,-174,769,-1119,945,-176,-943,1888,-2064,1121
%N A247917 Expansion of 1 / (1 + x - x^3) in powers of x.
%H A247917 Vincenzo Librandi, <a href="/A247917/b247917.txt">Table of n, a(n) for n = 0..1000</a>
%H A247917 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1).
%F A247917 G.f.: 1 / (1 + x - x^3).
%F A247917 0 = a(n) - a(n+2) - a(n+3) for all n in Z.
%F A247917 a(-n) = A000931(n) for all n in Z.
%F A247917 a(n) = A176971(n+3) for all n in Z.
%F A247917 -a(n) = A104769(n+1) for all n in Z.
%F A247917 (-1)^n * a(n) = A050935(n+3) for all n in Z.
%F A247917 -(-1)^n * a(n) = A078013(n+3) for all n in Z.
%e A247917 G.f. = 1 - x + x^2 - x^4 + 2*x^5 - 2*x^6 + x^7 + x^8 - 3*x^9 + 4*x^10 + ...
%t A247917 CoefficientList[Series[1/(1 + x - x^3), {x, 0, 100}], x] (* _Vincenzo Librandi_, Sep 27 2014 *)
%t A247917 LinearRecurrence[{-1,0,1},{1,-1,1},60] (* _Harvey P. Dale_, Apr 10 2025 *)
%o A247917 (PARI) {a(n) = if( n<0, n = -3-n; polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n), polcoeff( 1 / (1 + x - x^3) + x * O(x^n), n))};
%o A247917 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 + x - x^3)));  // _G. C. Greubel_, Aug 04 2018
%Y A247917 Cf. A000931, A050935, A078013, A104769, A176971.
%K A247917 sign,easy
%O A247917 0,6
%A A247917 _Michael Somos_, Sep 26 2014