A097316 Chebyshev U(n,x) polynomial evaluated at x=33.

1, 66, 4355, 287364, 18961669, 1251182790, 82559102471, 5447649580296, 359462313197065, 23719065021425994, 1565098829100918539, 103272803655639197580, 6814439942443086121741, 449649763397588044837326, 29670069944298367873141775, 1957774966560294691582519824

Offset

0,2

Comments

Used to form integer solutions of Pell equation a^2 - 17*b^2 =-1. See A078989 with A078988.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..548

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

Rigoberto Flórez, Robinson A. Higuita, and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.5.

Tanya Khovanova, Recursive Sequences.

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

Formula

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

a(n) = S(n, 66) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. See A049310.

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

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*66^(n-2*k).

a(n) = ((33+8*sqrt(17))^(n+1) - (33-8*sqrt(17))^(n+1))/(16*sqrt(17)).

From Amiram Eldar, Jan 05 2026: (Start)

Product_{n>=1} (1 - 1/a(n)) = (16+4*sqrt(17))/33.

Product_{n>=0} (1 + 1/a(n)) = 1 + sqrt(17)/4. (End)

Maple

seq( simplify(ChebyshevU(n, 33)), n=0..20); # G. C. Greubel, Dec 22 2019

Mathematica

LinearRecurrence[{66, -1}, {1, 66}, 14] (* Ray Chandler, Aug 11 2015 *)
ChebyshevU[Range[21] -1, 33] (* G. C. Greubel, Dec 22 2019 *)

Prog

(PARI) vector( 21, n, polchebyshev(n-1, 2, 33) ) \\ G. C. Greubel, Dec 22 2019
(Magma) m:=33; I:=[1, 2*m]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 22 2019
(SageMath) [chebyshev_U(n, 33) for n in (0..20)] # G. C. Greubel, Dec 22 2019
(GAP) m:=33;; a:=[1, 2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019

Crossrefs

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), this sequence (m=33).

Cf. A049310, A078988, A078989.

Sequence in context: A004998 A239409 A295592 * A239337 A099639 A003555

Adjacent sequences: A097313 A097314 A097315 * A097317 A097318 A097319

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved