A086926 Product of Fibonacci and (shifted) triangular numbers.

0, 0, 1, 6, 18, 50, 120, 273, 588, 1224, 2475, 4895, 9504, 18174, 34307, 64050, 118440, 217192, 395352, 714951, 1285350, 2298660, 4091241, 7250221, 12797568, 22507500, 39452725, 68942718, 120132558, 208776974, 361937400, 626015085, 1080441264

Offset: 0

Author

James FitzSimons (cherry(AT)getnet.net), Sep 20 2003

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Crossrefs

Cf. A000045, A000217, A045925.

Programs

Mathematica

Array[Fibonacci[#] PolygonalNumber[# - 1] &, 33, 0] (* or *)
LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 1, 6, 18, 50}, 33] (* or *)
CoefficientList[Series[x^2*(1 + 3 x + x^3)/(1 - x - x^2)^3, {x, 0, 32}], x] (* Michael De Vlieger, Dec 17 2017 *)

MuPAD

numlib::fibonacci(n)*binomial(n,2) $ n = 0..35; // Zerinvary Lajos, May 09 2008

PARI

concat(vector(2), Vec(x^2*(1 + 3*x + x^3) / (1 - x - x^2)^3 + O(x^40))) \\ Colin Barker, Sep 20 2017

Formula

From Franklin T. Adams-Watters, Feb 03 2006: (Start)

a(n) = A000045(n)*A000217(n-1) = A000045(n)*n*(n-1)/2.

a(n) = (n/(n-2)*a(n-1) + n*(n-1))/((n-2)*(n-3))*a(n-2).

G.f.: x^2*(1+3x+x^3)/(1-x-x^2)^3. (End)

a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} i * C(n-k-1,k). - Wesley Ivan Hurt, Sep 19 2017

From Colin Barker, Sep 20 2017: (Start)

a(n) = ((-1)*(2^(-1-n)*((1-sqrt(5))^n - (1+sqrt(5))^n)*(-1+n)*n)) / sqrt(5).

a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>5. (End)

a(n) = A001629(n-2)+3*A001629(n-3)+A001629(n-5). - R. J. Mathar, May 16 2025

Extensions

Definition and more terms from Franklin T. Adams-Watters, Feb 03 2006