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%I #7 Sep 08 2022 08:45:07
%S 3,1,0,6,11,15,28,56,103,185,340,630,1159,2127,3912,7200,13243,24353,
%T 44792,82390,151539,278719,512644,942904,1734271,3189817,5866988,
%U 10791078,19847887,36505951,67144912,123498752,227149619,417793281
%N a(n) = Sum_{j=0..floor(n/2)} (-1)^(j+floor(n/2))*S(2j+q), where S(n) are generalized tribonacci numbers (A001644) and q = (1-(-1)^n)/2.
%C a(n) is the convolution of S(n) with the sequence (1,0,-1,0,1,0,-1,0,....) A056594.
%H G. C. Greubel, <a href="/A074678/b074678.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,1,1).
%F a(n) = Sum_{j=0..floor(n/2)} (-1)^(j+floor(n/2))*S(2j+q), where S(n) are generalized tribonacci numbers (A001644) and q = (1-(-1)^n)/2.
%F a(n) = a(n-1) + 2*a(n-3) + a(n-4) + a(n-5), a(0)=3, a(1)=1, a(2)=0, a(3)=6, a(4)=11.
%F G.f.: (3 - 2*x - x^2)/(1 - x - 2*x^3 - x^4 - x^5).
%t CoefficientList[Series[(3-2*x-x^2)/(1-x-2*x^3-x^4-x^5), {x, 0, 40}], x]
%t LinearRecurrence[{1,0,2,1,1}, {3,1,0,6,11}, 40] (* _G. C. Greubel_, Apr 02 2019 *)
%o (PARI) my(x='x+O('x^40)); Vec((3-2*x-x^2)/(1-x-2*x^3-x^4-x^5)) \\ _G. C. Greubel_, Apr 02 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (3-2*x-x^2)/(1-x-2*x^3-x^4-x^5) )); // _G. C. Greubel_, Apr 02 2019
%o (Sage) ((3-2*x-x^2)/(1-x-2*x^3-x^4-x^5)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 02 2019
%o (GAP) a:=[3,1,0,6,11];; for n in [6..40] do a[n]:=a[n-1]+2*a[n-3]+a[n-4] +a[n-5]; od; a; # _G. C. Greubel_, Apr 02 2019
%Y Cf. A001644, A056594.
%K easy,nonn
%O 0,1
%A Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2002
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