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%I #24 Aug 08 2022 09:40:33
%S 1,1,0,0,0,1,1,0,1,0,0,1,0,2,0,-1,2,-1,2,1,-2,4,-3,1,4,-5,7,-4,-2,10,
%T -12,11,-1,-11,22,-23,13,11,-33,45,-35,3,44,-78,81,-37,-41,122,-158,
%U 119,4,-163,281,-276,115,167,-443,558,-391,-52,611,-1000,949
%N Expansion of (1 + x) / ((1 - x^4) * (1 + x^4 - x^5)) in powers of x.
%H Vincenzo Librandi, <a href="/A247918/b247918.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1,2,-2,2,-1).
%F G.f.: 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 + x - x^3)).
%F a(n) = a(n+1) + a(n+5) - mod(floor((n-1)/2), 2) for all n in Z.
%F a(n) = -A247907(-8-n) for all n in Z.
%F Convolution of A077905 and A112553.
%F a(n) = (35 + 7*(A056594(n) + 3*A056594(n-1)) + 10*(3*A010892(n) - A010892(n-1)) - 2*(A176971(n) + 4*A172971(n-1) + 12*A176971(n-2)))/70. - _G. C. Greubel_, Aug 08 2022
%e G.f. = 1 + x + x^5 + x^6 + x^8 + x^11 + 2*x^13 - x^15 + 2*x^16 - x^17 + ...
%t CoefficientList[Series[(1+x)/((1-x^4)(1+x^4-x^5)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Sep 27 2014 *)
%o (PARI) {a(n) = if( n<0, n=-8-n; polcoeff( -1/((1-x)*(1-x+x^2)*(1+x^2)*(1 - x^2 - x^3)) + x * O(x^n), n), polcoeff( 1/((1-x)*(1-x+x^2)*(1+x^2)*(1+x-x^3)) + x * O(x^n), n))};
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!((1 + x)/((1-x^4)*(1+x^4-x^5)))); // _G. C. Greubel_, Aug 04 2018
%o (SageMath)
%o def A247918_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+x)/((1-x^4)*(1+x^4-x^5)) ).list()
%o A247918_list(70) # _G. C. Greubel_, Aug 08 2022
%Y Cf. A010892, A056594, A077905, A112553, A176971, A247907.
%K sign,easy
%O 0,14
%A _Michael Somos_, Sep 26 2014
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