A337284 a(n) = Sum_{i=1..n} (i-1)*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

0, 1, 3, 15, 79, 324, 1338, 5370, 20858, 79907, 301917, 1127753, 4175945, 15347222, 56045572, 203563012, 735880196, 2649245173, 9502874215, 33976624115, 121128306995, 430701953720, 1527852568478, 5408197139806, 19106052817630, 67376379676855, 237205619596129, 833831061604429, 2926954896983117

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

Formula

Schumacher (on page 194) gives two explicit formulas for a(n) in terms of tribonacci numbers.

From Colin Barker, Sep 14 2020: (Start)

G.f.: x^2*(1 - 2*x + 2*x^2 + 12*x^3 + 8*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 2*x^9) / ((1 - x)*(1 + x + x^2 - x^3)^2*(1 - 3*x - x^2 - x^3)^2)

a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3) - 35*a(n-4) + 3*a(n-5) + 48*a(n-7) - 11*a(n-8) + 7*a(n-9) - 14*a(n-10) + 2*a(n-11) - a(n-12) + a(n-13) for n>13.

(End)

a(n) = A337283(n) - A107239(n). - 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]]];
a[n_]:= a[n]= Sum[(j-1)*T[j]^2, {j, 0, n}];
Table[a[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+2*x^2+12*x^3+8*x^5+2*x^6+4*x^7+3*x^8+2*x^9)/((1-x)*(1-2*x-3*x^2-6*x^3+x^4+x^6)^2) )); // 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 A337284(n): return sum( (j-1)*T(j)^2 for j in (0..n) )
[A337284(n) for n in (1..40)] # G. C. Greubel, Nov 22 2021

Crossrefs

Cf. A000073, A085697, A107239, A337282, A337283, A337285.

Sequence in context: A298576 A299568 A031884 * A186942 A193913 A292097

Adjacent sequences: A337281 A337282 A337283 * A337285 A337286 A337287

Keyword

nonn

Author

N. J. A. Sloane, Sep 12 2020

Status

approved