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 A068203 Chebyshev T-polynomials T(n,15) with Diophantine property. 4
 1, 15, 449, 13455, 403201, 12082575, 362074049, 10850138895, 325142092801, 9743412645135, 291977237261249, 8749573705192335, 262195233918508801, 7857107443850071695, 235451028081583642049, 7055673735003659189775, 211434761022028192051201 (list; graph; refs; listen; history; text; internal format)
 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 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 xrange(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

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Last modified October 16 23:12 EDT 2018. Contains 316275 sequences. (Running on oeis4.)