0,2

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

Prime values: a(2) = 449, a(8) = 325142092801; Semiprimes: a(1) = 15 = 3 * 5, a(4) = 403201 = 191 * 2111. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 17 2005

H. W. Lenstra Jr., Solving the Pell Equation, Notices Amer. Math. Soc., 49 (No. 2, 2002), 182-192.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Zerinvary Lajos, Sage Notebooks

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 (vladeta(AT)Eunet.yu), 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 (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 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 (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004

a(n) = Cosh[2n*ArcSinh[Sqrt[7]]] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008

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

sage: [lucas_number2(n, 30, 1)/2 for n in xrange(0, 15)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

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

Sequence in context: A034675 A069431 A133791 this_sequence A020285 A041423 A041420

Adjacent sequences: A068200 A068201 A068202 this_sequence A068204 A068205 A068206

nonn,easy

njas, Mar 24 2002

More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de) and Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 25 2002