The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A097313 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=15. 21
 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 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 Table[GegenbauerC[n, 1, 15], {n, -1, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *) LinearRecurrence[{30, -1}, {0, 1}, 50] (* Vincenzo Librandi, Feb 12 2012 *) ChebyshevU[Range[22] -2, 15] (* G. C. Greubel, Dec 22 2019 *) PROG (Sage) [lucas_number1(n, 30, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 27 2008 (Sage) [chebyshev_U(n, 15) for n in (-1..20)] # G. C. Greubel, Dec 22 2019 (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 (PARI) vector( 21, n, polchebyshev(n-2, 2, 15) ) \\ G. C. Greubel, Dec 22 2019 (GAP) m:=15;; a:=[0, 1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019 CROSSREFS Cf. A200442. Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), this sequence (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33). Sequence in context: A218732 A158580 A171335 * A056389 A056379 A320670 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
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 23 08:11 EST 2021. Contains 340385 sequences. (Running on oeis4.)