A337286 a(n) = Sum_{i=0..n} i^2*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

0, 0, 4, 13, 77, 477, 2241, 10522, 47386, 204202, 860302, 3546623, 14357567, 57286271, 225714755, 879795380, 3397426356, 13012405492, 49478890936, 186932228945, 702169068945, 2623863676449, 9758799153349, 36140284390030, 133317609306766, 490032600916766, 1795262239190210, 6557012850772931

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. (Note that this paper uses an offset for the tribonacci numbers that is different from that used in A000073.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7,-9,-7,-56,96,108,252,-162,-114,-318,126,-16,136,-36,12,-21,3,-1,1).

Formula

G.f.: x^2*(4 - 15*x + 22*x^2 + 83*x^3 - 90*x^4 + 11*x^5 - 128*x^6 - 207*x^7 - 224*x^8 - 233*x^9 - 162*x^10 - 147*x^11 - 58*x^12 - 3*x^13 - 4*x^14 - x^15)/((1-x)*(1 + x + x^2 - x^3)^3*(1 - 3*x - x^2 - x^3)^3). - G. C. Greubel, Nov 22 2021

Mathematica

T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
A337286[n_]:= Sum[j^2*T[j]^2, {j, 0, n}];
Table[A337286[n], {n, 0, 50}] (* G. C. Greubel, Nov 22 2021 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0, 0] cat Coefficients(R!( x^2*(4-15*x+22*x^2+83*x^3-90*x^4+11*x^5-128*x^6-207*x^7-224*x^8-233*x^9-162*x^10- 147*x^11-58*x^12-3*x^13-4*x^14-x^15)/((1-x)*(1+x+x^2-x^3)^3*(1-3*x-x^2-x^3)^3) )); // G. C. Greubel, Nov 22 2021
(Sage)
@CachedFunction
def T(n): # A000073
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A337286(n): return sum( j^2*T(j)^2 for j in (0..n) )
[A337286(n) for n in (0..40)] # G. C. Greubel, Nov 22 2021

Crossrefs

Cf. A000073, A085697, A337282, A337283, A337284, A337285.

Sequence in context: A012261 A012075 A197942 * A140417 A301396 A221512

Adjacent sequences: A337283 A337284 A337285 * A337287 A337288 A337289

Keyword

nonn

Author

N. J. A. Sloane, Sep 12 2020

Status

approved