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A001582
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Product of Fibonacci and Pell numbers.
(Formerly M1966 N0779)
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4
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1, 2, 10, 36, 145, 560, 2197, 8568, 33490, 130790, 510949, 1995840, 7796413, 30454814, 118965250, 464711184, 1815292333, 7091038640, 27699580729, 108202305420, 422668460890, 1651061182538, 6449506621417, 25193576136960
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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J. L. Diaz-Barrero and J. J. Egozcue, Problem H-605, Fib. Q., 43 (No. 1, 2005), 92.
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FORMULA
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G.f.: (1-x^2)/(1-2*x-7*x^2-2*x^3+x^4).
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
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MAPLE
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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]
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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