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A164300
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a(n) = ((1+4*sqrt(2))*(4+sqrt(2))^n + (1-4*sqrt(2))*(4-sqrt(2))^n)/2.
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8
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1, 12, 82, 488, 2756, 15216, 83144, 452128, 2453008, 13294272, 72012064, 389976704, 2111644736, 11433484032, 61904845952, 335169991168, 1814692086016, 9825156811776, 53195565289984, 288012326955008
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OFFSET
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0,2
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COMMENTS
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Binomial transform of A164299. Fourth binomial transform of A164587. Inverse binomial transform of A164301.
This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 -2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021
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LINKS
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FORMULA
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a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 12.
G.f.: (1+4*x)/(1-8*x+14*x^2).
E.g.f.: (4*sqrt(2)*sinh(sqrt(2)*x) + cosh(sqrt(2)*x))*exp(4*x). - Ilya Gutkovskiy, Jun 24 2016
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*3^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)
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MATHEMATICA
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LinearRecurrence[{8, -14}, {1, 12}, 30] (* Harvey P. Dale, Apr 13 2012 *)
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PROG
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(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(4+r)^n+(1-4*r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009
(PARI) my(x='x+O('x^50)); Vec((1+4*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Sep 13 2017
(Sage) [( (1+4*x)/(1-8*x+14*x^2) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009
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EXTENSIONS
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STATUS
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approved
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