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A088317
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a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.
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4
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1, 4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236, 782369683393860737, 6355271576489378132, 51624542295308885793
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = ( (4+sqrt(17))^n + (4-sqrt(17))^n )/2.
Lim_{n -> oo} a(n+1)/a(n) = 4 + sqrt(17).
E.g.f.: exp(4*x)*cosh(sqrt(17)*x).
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*17^k*4^(n-2*k).
a(n) = (-i)^n * T(n, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
a(n) = (1/2)*((33+8*sqrt(17))*(4-sqrt(17))^(n+2) + (33-8*sqrt(17))*(4+sqrt(17))^(n+2)). - Harvey P. Dale, May 07 2012
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MATHEMATICA
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LinearRecurrence[{8, 1}, {1, 4}, 30] (* or *) With[{c=Sqrt[17]}, Simplify/@ Table[1/2 (c-4)((c+4)^n-(4-c)^n (33+8c)), {n, 30}]] (* Harvey P. Dale, May 07 2012 *)
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PROG
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(Maxima)
a[0]:1$ a[1]:4$ a[n]:=8*a[n-1]+a[n-2]$ A088317(n):=a[n]$
(Magma) [n le 2 select 4^(n-1) else 8*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022
(SageMath)
A088317=BinaryRecurrenceSequence(8, 1, 1, 4)
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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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Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003
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STATUS
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approved
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