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%I M1966 N0779 #92 Aug 05 2023 11:46:17
%S 1,2,10,36,145,560,2197,8568,33490,130790,510949,1995840,7796413,
%T 30454814,118965250,464711184,1815292333,7091038640,27699580729,
%U 108202305420,422668460890,1651061182538,6449506621417,25193576136960
%N Product of Fibonacci and Pell numbers.
%C Also number of perfect matchings (or domino tilings) in the graph W_4 X P_n.
%C 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
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001582/b001582.txt">Table of n, a(n) for n = 0..200</a>
%H J. L. Diaz-Barrero and J. J. Egozcue, <a href="http://www.fq.math.ca/Problems/advanced43-1.pdf">Problem H-605</a>, Fib. Q., 43 (No. 1, 2005), 92.
%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>
%H Ira M. Gessel and Ishan Kar, <a href="https://arxiv.org/abs/2304.10426">Binomial convolutions for rational power series</a>, arXiv:2304.10426 [math.CO], 2023.
%H D. C. Mead, <a href="http://www.fq.math.ca/Scanned/3-3/mead.pdf">An elementary method of summation</a>, Fib. Quart. 3 (1965), 209-213.
%H I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mezo/mezo5.html">Several Generating Functions for Second-Order Recurrence Sequences </a>, JIS 12 (2009) 09.3.7.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Sellers/sellers4.html">Domino Tilings and Products of Fibonacci and Pell Numbers</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HoradamSequence.html">Horadam Sequence</a>.
%H Yifan Zhang and George Grossman, <a href="https://www.emis.de/journals/JIS/VOL21/Zhang/zhang44.html">A Combinatorial Proof for the Generating Function of Powers of a Second-Order Recurrence Sequence</a>, J. Int. Seq. 21 (2018), #18.3.3.
%H <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,2,-1).
%F G.f.: (1-x^2)/(1-2*x-7*x^2-2*x^3+x^4).
%F From _Kieren MacMillan_, Sep 29 2008: (Start)
%F a(n) = 11*a(n-2) + 16*a(n-3) + 3*a(n-4) - 2*a(n-5).
%F a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4). (End)
%F 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
%F a(n) = A166989(n) - A166989(n-2). - _R. J. Mathar_, Jul 14 2016
%p 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]
%t 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 *)
%Y Cf. A000045, A000129.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 01 2000
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