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

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

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

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.
• R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (66,-1).

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).

Programs

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

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

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 *)

PARI

vector( 21, n, polchebyshev(n-1, 2, 33) ) \\ G. C. Greubel, Dec 22 2019

Sage

[chebyshev_U(n,33) for n in (0..20)] # G. C. Greubel, Dec 22 2019

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)).