A337283 a(n) = Sum_{i=0..n} i*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

0, 0, 2, 5, 21, 101, 395, 1578, 6186, 23610, 89220, 333431, 1234343, 4536551, 16567157, 60172532, 217524468, 783111476, 2809027334, 10043413545, 35805255545, 127314522569, 451629771519, 1598650868766, 5647706073630, 19916305738030, 70117445671624, 246478579433947, 865201260035147

Offset

0,3

References

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-2,-2,-35,3,0,48,-11,7,-14,2,-1,1).

Formula

From Colin Barker, Sep 13 2020: (Start)

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

(End)

a(n) = Sum_{j=0..n} j*A085697(j). - G. C. Greubel, Nov 20 2021

Mathematica

T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
a[n_]:= a[n]= Sum[j*T[j]^2, {j, 0, n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 20 2021 *)

Prog

(PARI) concat([0, 0], Vec(x^2*(1+x)*(2 -7*x +7*x^2 +3*x^3 +9*x^4 +7*x^5 +x^6 + x^7 +x^8)/((1-x)*(1 +x +x^2 -x^3)^2*(1 -3*x -x^2 -x^3)^2) + O(x^30))) \\ Colin Barker, Sep 19 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0, 0] cat Coefficients(R!( x^2*(1-x^2)*(2-7*x+7*x^2+3*x^3+9*x^4+7*x^5+x^6+x^7+x^8)/((1-x)*(1+x+x^2-x^3)*(1-3*x-x^2-x^3))^2 )); // G. C. Greubel, Nov 20 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 A337283(n): return sum( j*T(j)^2 for j in (0..n) )
[A337283(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021

Crossrefs

Cf. A000073, A085697, A337282, A337284.

Sequence in context: A319850 A270597 A174080 * A115087 A008981 A008982

Adjacent sequences: A337280 A337281 A337282 * A337284 A337285 A337286

Keyword

nonn,easy

Author

N. J. A. Sloane, Sep 12 2020

Status

approved