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A029548
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Expansion of 1/(1-32*x+x^2).
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3
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1, 32, 1023, 32704, 1045505, 33423456, 1068505087, 34158739328, 1092011153409, 34910198169760, 1116034330278911, 35678188370755392, 1140585993533893633, 36463073604713840864, 1165677769357309014015
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution from Bruno Berselli, Nov 21 2011: (Start)
A Diophantine property of these numbers: ((a(n+1)-a(n-1))/2)^2 - 255*a(n)^2 = 1.
More generally, for t(m)=m+sqrt(m^2-1) and u(n)=(t(m)^(n+1)-1/t(m)^(n+1))/(t(m)-1/t(m)), we can verify that ((u(n+1)-u(n-1))/2)^2-(m^2-1)*u(n)^2=1. (End)
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients, signature (32,-1).
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FORMULA
| a(n) = 32*a(n-1) - a(n-2), a(-1)=0, a(0)=1.
a(n) = S(n, 32) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310. - Wolfdieter Lang, Nov 29 2002
a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap=16+sqrt(255) and am=16-sqrt(255).
a(n) = sum((-1)^k*binomial(n-k, k)*32^(n-2*k), k=0..floor(n/2)).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*31^k. - DELEHAM Philippe, Feb 10 2012
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MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 16]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky, Sep 11 2008]
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PROG
| (Other) sage: [lucas_number1(n, 32, 1) for n in xrange(1, 16)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 07 2009]
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CROSSREFS
| Cf. A200441, A200442.
Sequence in context: A065552 A158617 A171337 * A016745 A189267 A189620
Adjacent sequences: A029545 A029546 A029547 * A029549 A029550 A029551
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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