A097844 Chebyshev polynomials S(n,171).

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

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A097845, A098244.

Programs

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

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-171*x+x^2) )); // G. C. Greubel, Jan 14 2019

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 *)

PARI

my(x='x+O('x^30)); Vec(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

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).