A097311 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=14.

0, 1, 28, 783, 21896, 612305, 17122644, 478821727, 13389885712, 374437978209, 10470873504140, 292810020137711, 8188209690351768, 228977061309711793, 6403169506981578436, 179059769134174484415, 5007270366249903985184

Offset

-1,3

Comments

b(n)^2 - 195*a(n)^2 = +1 with b(n):=A097310(n) gives all nonnegative integer solutions of this Pell equation.

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 28'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

For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,27}. - Milan Janjic, Jan 26 2015

Links

Indranil Ghosh, Table of n, a(n) for n = -1..689

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (28,-1).

Formula

a(n) = S(n, 28) = U(n, 14), n>=-1, with Chebyshev polynomials of the second kind. See A049310 for the triangle of S(n, x) coefficients. S(-1, x) := 0 =: U(-1, x).

G.f.: 1/(1-28*x+x^2).

a(n) = ((14+sqrt(195))^(n+1) - (14-sqrt(195))^(n+1))/(2*sqrt(195)), (Binet form).

a(n) = 28*a(n-1)-a(n-2); a(0)=1, a(1)=28; a(-1)=0. - Zerinvary Lajos, Apr 29 2009

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*27^k. - Philippe Deléham, Feb 10 2012

With an offset of 0, product {n >= 1} (1 + 1/a(n)) = 1/13*(13 + sqrt(195)). - Peter Bala, Dec 23 2012

Product {n >= 2} (1 - 1/a(n)) = 1/28*(13 + sqrt(195)). - Peter Bala, Dec 23 2012

a(n) = sqrt((A097310(n)^2 - 1)/195).

Maple

seq( simplify(ChebyshevU(n, 14)), n=-1..20); # G. C. Greubel, Dec 23 2019

Mathematica

Table[GegenbauerC[n, 1, 14], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
LinearRecurrence[{28, -1}, {0, 1}, 17] (* Ray Chandler, Aug 12 2015 *)
ChebyshevU[Range[21] -2, 14] (* G. C. Greubel, Dec 23 2019 *)

Prog

(Sage) [lucas_number1(n, 28, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 25 2008
(Sage) [chebyshev_U(n, 14) for n in (-1..20)] # G. C. Greubel, Dec 23 2019
(PARI) vector( 22, n, polchebyshev(n-2, 2, 14) ) \\ G. C. Greubel, Dec 23 2019
(Magma) m:=14; I:=[0, 1]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 23 2019
(GAP) m:=14;; 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 23 2019

Crossrefs

Cf. A049310, A097310.

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), this sequence (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).

Sequence in context: A158545 A291997 A171333 * A223495 A209228 A208505

Adjacent sequences: A097308 A097309 A097310 * A097312 A097313 A097314

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved