A337285 a(n) = Sum_{i=1..n} (i-1)^2*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

0, 1, 5, 41, 297, 1522, 7606, 35830, 159734, 691175, 2911275, 11995471, 48573775, 193800376, 763577276, 2976338876, 11493413820, 44020618429, 167385941185, 632387189285, 2375420846885, 8876467428110, 33013780952786, 122261706093330, 451010242361106, 1657768413841731, 6073328651742855

Offset

1,3

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 = 1..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

From G. C. Greubel, Nov 22 2021: (Start)

a(n) = A337286(n) - 2*A337283(n) + A107239(n).

a(n) = Sum_{j=0..n-1} j^2*A000073(j+1)^2.

G.f.: x^2*(1 -2*x +15*x^2 +62*x^3 -97*x^4 +96*x^5 +73*x^6 -64*x^7 -57*x^8 -194*x^9 -127*x^10 -138*x^11 -55*x^12 -12*x^13 -9*x^14 -4*x^15)/((1-x)*(1 +x +x^2 -x^3)^3*(1 -3*x -x^2 -x^3)^3). (End)

Mathematica

T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
A337285[n_]:= Sum[j^2*T[j+1]^2, {j, 0, n-1}];
Table[A337285[n], {n, 40}] (* G. C. Greubel, Nov 22 2021 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1 -2*x+15*x^2+62*x^3-97*x^4+96*x^5+73*x^6-64*x^7-57*x^8-194*x^9-127*x^10-138*x^11 -55*x^12-12*x^13-9*x^14-4*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 A337285(n): return sum( j^2*T(j+1)^2 for j in (0..n-1) )
[A337285(n) for n in (1..40)] # G. C. Greubel, Nov 22 2021

Crossrefs

Cf. A000073, A085697, A107239, A337282, A337283, A337284, A337286.

Sequence in context: A218349 A154577 A198688 * A349541 A202249 A067381

Adjacent sequences: A337282 A337283 A337284 * A337286 A337287 A337288

Keyword

nonn

Author

N. J. A. Sloane, Sep 12 2020

Status

approved