login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 23 13:09 EST 2017. Contains 295127 sequences.