A097844 Chebyshev polynomials S(n,171).

1, 171, 29240, 4999869, 854948359, 146191169520, 24997835039561, 4274483600595411, 730911697866775720, 124981625851618052709, 21371127108928820237519, 3654337754000976642563040, 624870384807058077058042321, 106849181464252930200282673851

Offset

0,2

Comments

Used for all positive integer solutions of Pell equation x^2 - 173*y^2 = -4. See A097845 with A098244.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..446

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

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (171,-1).

Formula

a(n) = S(n, 171) = U(n, 171/2) = S(2*n+1, sqrt(173))/sqrt(173) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x).

a(n) = 171*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=171.

a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (171+13*sqrt(173))/2 and am = (171-13*sqrt(173))/2 = 1/ap.

G.f.: 1/(1-171*x+x^2).

Mathematica

CoefficientList[Series[1/(1-171x+x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{171, -1}, {1, 171}, 30] (* Harvey P. Dale, Mar 21 2013 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(1/(1-171*x+x^2)) \\ G. C. Greubel, Jan 14 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-171*x+x^2) )); // G. C. Greubel, Jan 14 2019
(Sage) (1/(1-171*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 14 2019
(GAP) a:=[1, 171];; for n in [3..30] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 14 2019

Crossrefs

Cf. A097845, A098244.

Sequence in context: A046166 A262113 A145625 * A076573 A015356 A259158

Adjacent sequences: A097841 A097842 A097843 * A097845 A097846 A097847

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 10 2004

Status

approved