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A164598
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a(n) = 12*a(n-1) - 34*a(n-2), for n > 1, with a(0) = 1, a(1) = 14.
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8
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1, 14, 134, 1132, 9028, 69848, 531224, 3999856, 29936656, 223244768, 1661090912, 12342768832, 91636134976, 679979479424, 5044125163904, 37410199666432, 277422140424448, 2057118896434688, 15253073982785024, 113094845314640896
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OFFSET
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0,2
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COMMENTS
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Binomial transform of A164301. Sixth binomial transform of A164587. Inverse binomial transform of A164599.
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 have 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 11 2021
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LINKS
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FORMULA
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a(n) = ((1+4*sqrt(2))*(6+sqrt(2))^n + (1-4*sqrt(2))*(6-sqrt(2))^n)/2.
G.f.: (1+2*x)/(1-12*x+34*x^2).
E.g.f.: exp(6*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*5^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)
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MAPLE
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m:=30; S:=series( (1+2*x)/(1-12*x+34*x^2), x, m+1):
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MATHEMATICA
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LinearRecurrence[{12, -34}, {1, 14}, 30] (* G. C. Greubel, Aug 11 2017 *)
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PROG
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(Magma) [ n le 2 select 13*n-12 else 12*Self(n-1)-34*Self(n-2): n in [1..30] ];
(PARI) my(x='x+O('x^30)); Vec((1+2*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 11 2017
(Sage) [( (1+2*x)/(1-12*x+34*x^2) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 11 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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STATUS
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approved
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