A001582 Product of Fibonacci and Pell numbers.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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.
• Thomas Koshy and Zhenguang Gao, Fibonacci-Pell hybridities, Int. J. Math. Educ. Sci. Technol., Vol. 43(6) (2012), 779-783.
• 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).

Crossrefs

Cf. A000045, A000129.

Programs

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 *)

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

Extensions

More terms from James Sellers, May 01 2000