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A368345 a(n) = Sum_{k=0..n} 4^(n-k) * floor(k/3).

%I A368345 #25 Jun 07 2025 16:52:43
%S A368345 0,0,0,1,5,21,86,346,1386,5547,22191,88767,355072,1420292,5681172,
%T A368345 22724693,90898777,363595113,1454380458,5817521838,23270087358,
%U A368345 93080349439,372321397763,1489285591059,5957142364244,23828569456984,95314277827944,381257111311785
%N A368345 a(n) = Sum_{k=0..n} 4^(n-k) * floor(k/3).
%H A368345 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4,1,-5,4).
%F A368345 a(n) = a(n-3) + (4^(n-2) - 1)/3.
%F A368345 a(n) = 1/3 * Sum_{k=0..n} floor(4^k/21) = Sum_{k=0..n} floor(4^k/63).
%F A368345 a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3) - 5*a(n-4) + 4*a(n-5).
%F A368345 G.f.: x^3/((1-x) * (1-4*x) * (1-x^3)).
%F A368345 a(n) = (floor(4^(n+1)/63) - floor((n+1)/3))/3.
%F A368345 E.g.f.: exp(-x/2)*(exp(3*x/2)*(4*exp(3*x) - 7 - 21*x) + 3*cos(sqrt(3)*x/2) + 9*sqrt(3)*sin(sqrt(3)*x/2))/189. - _Stefano Spezia_, Jun 07 2025
%o A368345 (PARI) a(n, m=3, k=4) = (k^(n+1)\(k^m-1)-(n+1)\m)/(k-1);
%Y A368345 Partial sums of A033140.
%Y A368345 Column k=4 of A368343.
%Y A368345 Cf. A097138.
%K A368345 nonn,easy
%O A368345 0,5
%A A368345 _Seiichi Manyama_, Dec 22 2023