A001582 Product of Fibonacci and Pell numbers. (Formerly M1966 N0779)

1, 2, 10, 36, 145, 560, 2197, 8568, 33490, 130790, 510949, 1995840, 7796413, 30454814, 118965250, 464711184, 1815292333, 7091038640, 27699580729, 108202305420, 422668460890, 1651061182538, 6449506621417, 25193576136960

Offset

0,2

Comments

Also number of perfect matchings (or domino tilings) in the graph W_4 X P_n.

In general, the termwise product of two Horadam sequences having signatures of (a,b) and (c,d) will be a fourth-order sequence with signature (a*c,a^2*d+2*b*d+b*c^2,a*b*c*d,-b^2*d^2). - Gary Detlefs, Oct 13 2020

a(n) + a(n-1) is the numerator of the continued fraction [1,...,1,2,...,2] with n 1's followed by n 2's. - Greg Dresden and Hexuan Wang, Aug 16 2021

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..200

J. L. Diaz-Barrero and J. J. Egozcue, Problem H-605, Fib. Q., 43 (No. 1, 2005), 92.

F. Faase, Counting Hamiltonian cycles in product graphs

F. Faase, Results from the counting program

Ira M. Gessel and Ishan Kar, Binomial convolutions for rational power series, arXiv:2304.10426 [math.CO], 2023.

D. C. Mead, An elementary method of summation, Fib. Quart. 3 (1965), 209-213.

I. Mezo, Several Generating Functions for Second-Order Recurrence Sequences , JIS 12 (2009) 09.3.7.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.

Eric Weisstein's World of Mathematics, Horadam Sequence.

Yifan Zhang and George Grossman, A Combinatorial Proof for the Generating Function of Powers of a Second-Order Recurrence Sequence, J. Int. Seq. 21 (2018), #18.3.3.

Index entries for sequences related to dominoes

Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Formula

G.f.: (1-x^2)/(1-2*x-7*x^2-2*x^3+x^4).

From Kieren MacMillan, Sep 29 2008: (Start)

a(n) = 11*a(n-2) + 16*a(n-3) + 3*a(n-4) - 2*a(n-5).

a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4). (End)

a(n) = ((10+5*sqrt(2)+2*sqrt(5)+sqrt(10))*((1+sqrt(2)+sqrt(5)+sqrt(10))/2)^n+(10-5*sqrt(2)-2*sqrt(5)+sqrt(10))*((1-sqrt(2)-sqrt(5)+sqrt(10))/2)^n+(10+5*sqrt(2)-2*sqrt(5)-sqrt(10))*((1+sqrt(2)-sqrt(5)-sqrt(10))/2)^n+(10-5*sqrt(2)+2*sqrt(5)-sqrt(10))*((1-sqrt(2)+sqrt(5)-sqrt(10))/2)^n)/40. - Tim Monahan, Aug 03 2011

a(n) = A166989(n) - A166989(n-2). - R. J. Mathar, Jul 14 2016

Maple

A001582:=-(z-1)*(1+z)/(1-2*z-7*z**2-2*z**3+z**4); # [conjectured (correctly) by Simon Plouffe in his 1992 dissertation]

Mathematica

CoefficientList[Series[(1-x^2)/(1-2x-7x^2-2x^3+x^4), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, 7, 2, -1}, {1, 2, 10, 36}, 30] (* Harvey P. Dale, May 01 2011 *)

Crossrefs

Cf. A000045, A000129.

Sequence in context: A192858 A202796 A335559 * A357035 A370713 A026546

Adjacent sequences: A001579 A001580 A001581 * A001583 A001584 A001585

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 01 2000

Status

approved