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%I #23 Sep 08 2022 08:45:08
%S 1,-1,1,1,-1,3,1,1,7,3,9,17,15,35,49,65,119,163,249,401,575,899,1377,
%T 2049,3175,4803,7273,11153,16879,25699,39185,59457,90583,137827,
%U 209497,318993,485151,737987,1123137,1708289,2599111,3954563,6015689,9152785,13924815
%N Expansion of (1-x)/(1-x^2-2*x^3).
%H G. C. Greubel, <a href="/A078026/b078026.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2).
%F a(n) = Sum_{m=1..n} Sum_{i=0..n-m} binomial(m+i-1,m-1)*Sum_{j=0..m} binomial(j,n-3*m+2*j-i)*2^(m-j)*binomial(m,j)*(-1)^(-n+3*m-j+i)). - _Vladimir Kruchinin_, May 12 2011
%p seq(coeff(series((1-x)/(-2*x^3-x^2+1), x, n+1), x, n), n = 0..50); # _G. C. Greubel_, Aug 04 2019
%t CoefficientList[Series[(1-x)/(1-x^2-2*x^3), {x,0,50}], x] (* _Wesley Ivan Hurt_, Jan 24 2017 *)
%o (Maxima)
%o a(n):=sum(sum(binomial(m+i-1,m-1)*sum(binomial(j,n-3*m+2*j-i)*2^(m-j)*binomial(m,j)*(-1)^(-n+3*m-j+i),j,0,m),i,0,n-m),m,1,n); /* _Vladimir Kruchinin_, May 12 2011 */
%o (PARI) Vec((1-x)/(1-x^2-2*x^3) + O(x^50)) \\ _Felix Fröhlich_, Jan 24 2017
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x^2-2*x^3) )); // _G. C. Greubel_, Aug 04 2019
%o (Sage) ((1-x)/(1-x^2-2*x^3)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 04 2019
%o (GAP) a:=[1,-1,1];; for n in [4..50] do a[n]:=a[n-2]+2*a[n-3]; od; a; # _G. C. Greubel_, Aug 04 2019
%K sign,easy
%O 0,6
%A _N. J. A. Sloane_, Nov 17 2002
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