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

Author

N. J. A. Sloane, Sep 12 2020

Keywords

• nonn

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).

Crossrefs

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

Programs

Magma

R:=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

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 *)

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

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)