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

Author

Wolfdieter Lang, Oct 18 2004

Keywords

• nonn
• easy

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

Also numbers k such that 17*(k-1)*(k+1) is a square. - Bruno Berselli, May 31 2025

Examples

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

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).

Crossrefs

Cf. A121470, A041024, A040012.

Row 4 of array A188645.

Programs

Mathematica

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

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

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).

Extensions

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