OFFSET
0,3
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1261
Hideyuki Ohtsuka, Problem H-736, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 51, No. 2 (2013), p. 186; On the Sum of the Cubes of the Tribonacci Numbers, Solution to Problem H-736 by the proposer, ibid., Vol. 53, No. 1 (2015), pp. 92-94.
Raphael Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, The Fibonacci Quarterly, Vol. 58, No. 3 (2020), pp. 194-202.
Index entries for linear recurrences with constant coefficients, signature (5,5,25,-58,26,-42,54,-13,1,-3,1).
FORMULA
a(n) = a(n-1) + (A000073(n))^3.
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 ).
a(n) = (1/22) * (T(n+3)^3 - 2 * T(n+2)^3 - 10 * T(n+1)^3 + T(n)^3 + 9 * T(n) * T(n+1)^2 + 3 * T(n)^2 * T(n+1) - 18 * T(n) * T(n+1) * T(n+2) + 3*n + 4), where T(n) = A000073(n+1) (Ohtsuka, 2015). - Amiram Eldar, Jan 10 2026
MATHEMATICA
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 *)
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 *)
PROG
(PARI) T(n)=([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073
a(n) = sum(k=1, n, T(k)^3); \\ Michel Marcus, Jan 12 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
R. J. Mathar, Dec 18 2013
STATUS
approved