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

0,2

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).

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A225492 A256194 A247141 * A267643 A267666 A020285

Adjacent sequences: A068200 A068201 A068202 * A068204 A068205 A068206

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 24 2002

Extensions

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

Additional term from Colin Barker, Feb 24 2014

Status

approved