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A228609 Partial sums of the cubes of the tribonacci sequence A000073.

%I A228609 #28 Jan 05 2025 19:51:40
%S A228609 0,1,2,10,74,417,2614,16438,101622,633063,3941012,24511836,152535900,
%T A228609 949133883,5905611508,36746590964,228646935796,1422699232325,
%U A228609 8852413871022,55082039340022,342734883853750,2132586518002125
%N A228609 Partial sums of the cubes of the tribonacci sequence A000073.
%D A228609 R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202.
%H A228609 Michael De Vlieger, <a href="/A228609/b228609.txt">Table of n, a(n) for n = 0..1261</a>
%H A228609 H. Ohtsuka, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/May2013advanced.pdf">Advanced Problems and Solutions</a>, Fib. Quart. 51 (2) (2013) 186, H-736.
%H A228609 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,25,-58,26,-42,54,-13,1,-3,1).
%F A228609 a(n) = a(n-1) + (A000073(n))^3.
%F A228609 G.f.: x*(-1+3*x+11*x^3-5*x^4+x^5-3*x^6+x^7+5*x^2) / ( (x^3-5*x^2+7*x-1) *(x^6+4*x^5+11*x^4+12*x^3+11*x^2+4*x+1) *(x-1)^2 )
%t A228609 CoefficientList[Series[x (-1 + 3 x + 11 x^3 - 5 x^4 + x^5 - 3 x^6 + x^7 + 5 x^2)/((x^3 - 5 x^2 + 7 x - 1) (x^6 + 4 x^5 + 11 x^4 + 12 x^3 + 11 x^2 + 4 x + 1) (x - 1)^2), {x, 0, 21}], x] (* _Michael De Vlieger_, Jan 12 2022 *)
%t A228609 Accumulate[LinearRecurrence[{1,1,1},{0,1,1},30]^3]  (* or *) LinearRecurrence[ {5,5,25,-58,26,-42,54,-13,1,-3,1},{0,1,2,10,74,417,2614,16438,101622,633063,3941012},30] (* _Harvey P. Dale_, Sep 11 2022 *)
%o A228609 (PARI) T(n)=([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073
%o A228609 a(n) = sum(k=1, n, T(k)^3); \\ _Michel Marcus_, Jan 12 2022
%Y A228609 Cf. A000073, A107239.
%K A228609 easy,nonn
%O A228609 0,3
%A A228609 _R. J. Mathar_, Dec 18 2013