A068203 Chebyshev T-polynomials T(n,15) with Diophantine property.

1, 15, 449, 13455, 403201, 12082575, 362074049, 10850138895, 325142092801, 9743412645135, 291977237261249, 8749573705192335, 262195233918508801, 7857107443850071695, 235451028081583642049, 7055673735003659189775, 211434761022028192051201

Offset: 0

Author

N. J. A. Sloane, Mar 24 2002

Keywords

• nonn
• easy

Comments

Let (x_n, y_n) be n-th solution to the Pell equation x^2 = 14*y^2 + 1. Sequence gives {x_n}.

Numbers n such that 14*(n^2-1) is a square. - Vincenzo Librandi, Aug 08 2010

Except for the first term, positive values of x (or y) satisfying x^2 - 30xy + y^2 + 224 = 0. - Colin Barker, Feb 24 2014

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..676
• Tanya Khovanova, Recursive Sequences
• H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (30,-1).

Crossrefs

a(n)=sqrt(1 + 224*A097313(n-1)^2), n>=0. Cf. A068204.

Programs

Maple

Digits := 1000: q := seq(floor(evalf(((15+4*sqrt(14))^n+(15-4*sqrt(14))^n)/2)+0.1), n=1..30);

Mathematica

a[0] = 1; a[1] = 15; a[n_] := 30a[n-1] - a[n-2]; Table[a[n], {n,0,16}] (* or *) LinearRecurrence[{30,-1}, {1,15}, 17] (* Indranil Ghosh, Feb 18 2017 *)

Sage

[lucas_number2(n,30,1)/2 for n in range(0,15)] # Zerinvary Lajos, Jun 27 2008

Formula

x_n + y_n*sqrt(14) = (x_1 + y_1*sqrt(14))^n.

a(n) = (-15/2-2*sqrt(14))*(-1/(-15-4*sqrt(14)))^n/(-15-4*sqrt(14))+(2*sqrt(14)-15/2)*(-1/(-15+4*sqrt(14)))^n/(-15+4*sqrt(14)). Recurrence: a(n) = 30*a(n-1)-a(n-2). G.f.: (1-15*x)/(1-30*x+x^2). - Vladeta Jovovic, Mar 25 2002

a(n) = T(n, 15)= (S(n, 30)-S(n-2, 30))/2 = S(n, 30)-15*S(n-1, 30) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp.second, kind. See A053120 and A049310. S(n, 30)=A097313(n). - Wolfdieter Lang, Aug 31 2004

a(n) = sum(((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*15)^(n-2*k), k=0..floor(n/2)), n>=1. - Wolfdieter Lang, Aug 31 2004

a(n) = cosh(2*n*arcsinh(sqrt(7))). - Herbert Kociemba, Apr 24 2008

Extensions

More terms from Sascha Kurz and Vladeta Jovovic, Mar 25 2002

Additional term from Colin Barker, Feb 24 2014