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

0, 1, 2, 10, 74, 417, 2614, 16438, 101622, 633063, 3941012, 24511836, 152535900, 949133883, 5905611508, 36746590964, 228646935796, 1422699232325, 8852413871022, 55082039340022, 342734883853750, 2132586518002125

Offset

0,3

References

R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1261

H. Ohtsuka, Advanced Problems and Solutions, Fib. Quart. 51 (2) (2013) 186, H-736.

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 )

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

Cf. A000073, A107239.

Sequence in context: A275550 A243250 A088189 * A001395 A047853 A151387

Adjacent sequences: A228606 A228607 A228608 * A228610 A228611 A228612

Keyword

easy,nonn

Author

R. J. Mathar, Dec 18 2013

Status

approved