A099368 Twice Chebyshev polynomials of the first kind, T(n,x), evaluated at x=51/2.

2, 51, 2599, 132498, 6754799, 344362251, 17555720002, 894997357851, 45627309530399, 2326097788692498, 118585359913786999, 6045527257814444451, 308203304788622880002, 15712323016961952435651, 801020270560270951338199

Offset: 0

Author

Wolfdieter Lang, Oct 18 2004

Keywords

• nonn
• easy

Comments

a(n) and b(n):= A097836(n-1) with b(0) = 0 are the improper and proper nonnegative solutions of the Pell equation a(n)^2 - 53*(7*b(n))^2 = +4. - Wolfdieter Lang, Jun 27 2013

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..584
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (51, -1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

Mathematica

LinearRecurrence[{51, -1}, {2, 51}, 15] (* or *) CoefficientList[Series[(2 - 51 x)/(1 - 51 x + x^2), {x, 0, 14}], x] (* Michael De Vlieger, Feb 08 2017 *)

Formula

a(n) = 51*a(n-1) - a(n-2), n >= 1; a(-1)=51, a(0)=2.

a(n) = S(n, 51) - S(n-2, 51) = 2*T(n, 51/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 51)=A097836(n). U-, resp. T-, are Chebyshev polynomials of the second, resp. first, case. See A049310 and A053120.

a(n)= ap^n + am^n, with ap:=(51 + 7*sqrt(53))/2 and am:=(51 - 7*sqrt(53))/2.

G.f.: (2-51*x)/(1-51*x+x^2).