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%I #26 Sep 08 2022 08:45:35
%S 1,1,3,5,12,25,54,116,249,535,1149,2468,5301,11386,24456,52529,112827,
%T 242341,520524,1118033,2401422,5158012,11078889,23796335,51112125,
%U 109783684,235804269,506483762,1087875984,2336647777,5018883507
%N a(1) = 1, a(n) = Sum_{k=1..n} (k mod 3) * a(n-k) for n >= 2.
%C Lim_{n -> infinity} a(n+1)/a(n) = 2.1478990357047874.
%H G. C. Greubel, <a href="/A141685/b141685.txt">Table of n, a(n) for n = 1..1001</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hessenberg_matrix">Hessenberg matrix</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,1).
%F a(n) = Sum_{k=1..n} (k mod 3) * a(n-k).
%F If p[i] = modp(i,3) and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i<=j), A[i,j] = -1, (i= j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n+1) = det A. - _Milan Janjic_, May 02 2010
%F G.f.: x*(1-x)*(1+x+x^2)/(1-x-2*x^2-x^3). - _Colin Barker_, Feb 01 2012
%t a[1]=1; a[n_]:= Sum[Mod[k, 3]*a[n-k], {k,1,n}]; Table[a[n], {n, 1, 35}]
%t Join[{1}, LinearRecurrence[{1,2,1}, {1,3,5}, 35]] (* _G. C. Greubel_, Apr 06 2019 *)
%o (PARI) my(x='x+O('x^35)); Vec(x*(1-x^3)/(1-x-2*x^2-x^3)) \\ _G. C. Greubel_, Apr 06 2019
%o (Magma) I:=[1,3,5]; [1] cat [n le 3 select I[n] else Self(n-1) +2*Self(n -2)+Self(n-3): n in [1..35]]; // _G. C. Greubel_, Apr 06 2019
%o (Sage) a=(x*(1-x^3)/(1-x-2*x^2-x^3)).series(x, 35).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Apr 06 2019
%K nonn
%O 1,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 08 2008
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