A099370 Chebyshev polynomial of the first kind, T(n,x), evaluated at x=33.

1, 33, 2177, 143649, 9478657, 625447713, 41270070401, 2723199198753, 179689877047297, 11856808685922849, 782369683393860737, 51624542295308885793, 3406437421806992601601, 224773245296966202819873

Offset

0,2

Comments

Used in A099369.

Solutions of the Pell equation x^2 - 17y^2 = 1 (x values). After initial term this sequence bisects A041024. See 8*A097316(n-1) with A097316(-1) = 0 for corresponding y values. a(n+1)/a(n) apparently converges to (4+sqrt(17))^2. (See related comments in A088317, which this sequence also bisects.). - Rick L. Shepherd, Jul 31 2006

From a(n) = T(n, 33) (see the formula section) and the de Moivre-Binet formula for T(n,x=33) follows a(n+1)/a(n) = 33 + 8*sqrt(17), which is the conjectured value (4+sqrt(17))^2 given in the previous comment by Rick L. Shepherd. - Wolfdieter Lang, Jun 28 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Pell Equation

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (66,-1).

Formula

a(n) = 66*a(n-1) - a(n-2), a(-1):= 33, a(0)=1.

a(n) = T(n, 33) = (S(n, 66)-S(n-2, 66))/2 = S(n, 66)-33*S(n-1, 66) with T(n, x), resp. S(n, x), Chebyshev polynomials of the first, resp.second, kind. See A053120 and A049310. S(n, 66)=A097316(n).

a(n) = ((33+8*sqrt(17))^n + (33-8*sqrt(17))^n)/2.

a(n) = Sum_{k=0..floor(n/2)} ((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*33)^(n-2*k), for n>=1, a(0)=1.

G.f.: (1-33*x)/(1-66*x+x^2).

Example

a(1)^2 - 17*A121470(1)^2 = 33^2 - 17*8^2 = 1089 - 1088 = 1.

Mathematica

LinearRecurrence[{66, -1}, {1, 33}, 14] (* Ray Chandler, Aug 11 2015 *)

Prog

(PARI) \\ Program uses fact that continued fraction for sqrt(17) = [4, 8, 8, ...].
print1("1, "); forstep(n=2, 40, 2, v=vector(n, i, if(i>1, 8, 4)); print1(contfracpnqn(v)[1, 1], ", ")) \\ Rick L. Shepherd, Jul 31 2006
(PARI) vector(20, n, polchebyshev(n-1, 1, 33)) \\ Joerg Arndt, Jan 01 2021

Crossrefs

Cf. A121470, A041024, A040012.

Row 4 of array A188645.

Sequence in context: A180920 A120288 A284112 * A267672 A283533 A294773

Adjacent sequences: A099367 A099368 A099369 * A099371 A099372 A099373

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 18 2004

Extensions

A-number for y values in Pell equation corrected by Wolfdieter Lang, Jun 28 2013

Status

approved