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A097313 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=15. 3
0, 1, 30, 899, 26940, 807301, 24192090, 724955399, 21724469880, 651009141001, 19508549760150, 584605483663499, 17518655960144820, 524975073320681101, 15731733543660288210, 471427031236487965199, 14127079203550978667760 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

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 n X n 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

For n>=0, a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,29}. - Milan Janjic, Jan 26 2015

LINKS

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 linear recurrences with constant coefficients, signature (30,-1).

FORMULA

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 Deléham, 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

MAPLE

0, seq(orthopoly[U](n, 15), n=0..50); # Robert Israel, Jan 26 2015

MATHEMATICA

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 15]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

LinearRecurrence[{30, -1}, {0, 1}, 50] (* Vincenzo Librandi, Feb 12 2012 *)

PROG

(Sage) [lucas_number1(n, 30, 1) for n in xrange(0, 20)] # Zerinvary Lajos, 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

CROSSREFS

Cf. A200442.

Sequence in context: A218732 A158580 A171335 * A056389 A056379 A171304

Adjacent sequences:  A097310 A097311 A097312 * A097314 A097315 A097316

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified March 24 22:16 EDT 2017. Contains 284035 sequences.