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A097313
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Chebyshev polynomials of the second kind, U(n,x), evaluated at x=15.
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3
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0, 1, 30, 899, 26940, 807301, 24192090, 724955399, 21724469880, 651009141001, 19508549760150, 584605483663499, 17518655960144820, 524975073320681101, 15731733543660288210, 471427031236487965199, 14127079203550978667760
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OFFSET
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-1,3
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COMMENTS
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b(n+1)^2 - 14*(4*a(n))^2 = +1, n>=-1, with b(n)=A068203(n) gives all nonnegative integer solutions of this D=224 Pell equation.
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 30's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
About first comment, 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. In this case is m=15. - Bruno Berselli, Nov 21 2011
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = -1..200
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients, signature (30,-1).
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FORMULA
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a(n)= S(n, 30) = U(n, 15), n>=-1, with Chebyshev polynomials of 2nd kind. See A049310 for the triangle of S(n, x) coefficients. S(-1, x) := 0 =: U(-1, x).
G.f.: 1/(1-30*x+x^2).
a(n)= ((15+4*sqrt(14))^(n+1) - (15-4*sqrt(14))^(n+1))/(8*sqrt(14)) (Binet form).
a(n) = sqrt((A068203(n+1)^2 - 1)/224), n>=-1.
a(n) = 30*a(n-1)-a(n-2) for n>0; a(-1)=0, a(0)=1. - Philippe DELEHAM, Nov 18 2008
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*29^k. - Philippe Deléham, Feb 10 2012
With an offset of 0, product {n >= 1} (1 + 1/a(n)) = 1/7*(7 + 2*sqrt(14)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/15*(7 + 2*sqrt(14)). - Peter Bala, Dec 23 2012
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MATHEMATICA
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lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 15]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky, Sep 11 2008]
LinearRecurrence[{30, -1}, {0, 1}, 50] (* Vincenzo Librandi, Feb 12 2012 *)
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PROG
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sage: [lucas_number1(n, 30, 1) for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008
(MAGMA) I:=[0, 1]; [n le 2 select I[n] else 30*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 12 2012
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CROSSREFS
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Cf. A200442.
Sequence in context: A218732 A158580 A171335 * A056389 A056379 A171304
Adjacent sequences: A097310 A097311 A097312 * A097314 A097315 A097316
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Aug 31 2004
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STATUS
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approved
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