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A097309 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=13. 21
0, 1, 26, 675, 17524, 454949, 11811150, 306634951, 7960697576, 206671502025, 5365498355074, 139296285729899, 3616337930622300, 93885489910449901, 2437406399741075126, 63278680903357503375, 1642808297087554012624 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

b(n)^2 - 42*(2*a(n))^2 = +1 with b(n):=A097308(n) gives all nonnegative integer solutions of this D:=42*4=168 Pell equation.

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 26'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,...,25}. Milan Janjic, Jan 25 2015

LINKS

Indranil Ghosh, Table of n, a(n) for n = -1..705

Tanya Khovanova, Recursive Sequences

Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (26,-1).

FORMULA

a(n) = S(n, 26) = U(n, 13), 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).

a(n) = ((13+2*sqrt(42))^n - (13-2*sqrt(42))^n)/(4*sqrt(42)), (Binet form).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-k, k) * 26^(n-2*k).

G.f.: 1/(1-26*x+x^2).

a(n) = 26*a(n-1) - a(n-2), a(-1)=0, a(0)=1. - Philippe Deléham, Nov 18 2008

a(n) = Sum_{k=0..n} A101950(n,k)*25^k. - Philippe Deléham, Feb 10 2012

With an offset of 0, product {n >= 1} (1 + 1/a(n)) = 1/6*(6 + sqrt(42)). - Peter Bala, Dec 23 2012

Product {n >= 2} (1 - 1/a(n)) = 1/13*(6 + sqrt(42)). - Peter Bala, Dec 23 2012

a(n) = sqrt((A097308(n)^2 - 1)/168).

MAPLE

seq( simplify(ChebyshevU(n, 13)), n=-1..20); # G. C. Greubel, Dec 22 2019

MATHEMATICA

Table[GegenbauerC[n, 1, 13], {n, -1, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

ChebyshevU[Range[22] -2, 13] (* G. C. Greubel, Dec 22 2019 *)

PROG

(Sage) [lucas_number1(n, 26, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 25 2008

(Sage) [chebyshev_U(n, 13) for n in (-1..20)] # G. C. Greubel, Dec 22 2019

(PARI) vector( 22, n, polchebyshev(n-2, 2, 13) ) \\ G. C. Greubel, Dec 22 2019

(MAGMA) m:=13; 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 22 2019

(GAP) m:=13;; 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. A097308, A101950.

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), this sequence (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).

Sequence in context: A209963 A158542 A171331 * A208778 A208499 A207587

Adjacent sequences:  A097306 A097307 A097308 * A097310 A097311 A097312

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified March 7 16:12 EST 2021. Contains 341897 sequences. (Running on oeis4.)