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

0,4

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).

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)

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 *)

Prog

(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

Crossrefs

Cf. A000045, A000217, A045925.

Sequence in context: A163765 A179754 A350155 * A328534 A003290 A318160

Adjacent sequences: A086923 A086924 A086925 * A086927 A086928 A086929

Keyword

nonn,easy

Author

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

Extensions

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

Status

approved