A097839 Chebyshev polynomials S(n,83).

1, 83, 6888, 571621, 47437655, 3936753744, 326703123097, 27112422463307, 2250004361331384, 186723249568041565, 15495779709786118511, 1285962992662679794848, 106719432611292636853873, 8856426943744626179076611, 734976716898192680226504840

Offset

0,2

Comments

Used for all positive integer solutions of Pell equation x^2 - 85*y^2 = -4. See A097840 with A097841.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..520

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 sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (83,-1).

Formula

a(n) = S(n, 83) = U(n, 83/2) = S(2*n+1, sqrt(85))/sqrt(85) 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) = 83*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=83.

a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap = (83+9*sqrt(85))/2 and am = (83-9*sqrt(85))/2 = 1/ap.

G.f.: 1/(1-83*x+x^2).

Mathematica

CoefficientList[Series[1/(1-83x+x^2), {x, 0, 20}], x] (* or *) LinearRecurrence[{83, -1}, {1, 83}, 20] (* Harvey P. Dale, Oct 11 2012 *)

Prog

(PARI) my(x='x+O('x^20)); Vec(1/(1-83*x+x^2)) \\ G. C. Greubel, Jan 13 2019
(Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-83*x+x^2) )); // G. C. Greubel, Jan 13 2019
(Sage) (1/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 13 2019
(GAP) a:=[1, 83];; for n in [3..20] do a[n]:=83*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2019

Crossrefs

Sequence in context: A252812 A202657 A180846 * A268987 A087189 A201727

Adjacent sequences: A097836 A097837 A097838 * A097840 A097841 A097842

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 10 2004

Extensions

More terms from Harvey P. Dale, Oct 11 2012

Status

approved