A097845 Chebyshev polynomials S(n,171) + S(n-1,171) with Diophantine property.

1, 172, 29411, 5029109, 859948228, 147046117879, 25144026209081, 4299481435634972, 735186181467371131, 125712537549484828429, 21496108734780438290228, 3675708881109905462800559

Offset

0,2

Comments

(13*a(n))^2 - 173*b(n)^2 = -4 with b(n) = A098244(n) give all positive solutions of this Pell equation.

Links

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

Tanya Khovanova, Recursive Sequences

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.

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) + S(n-1, 171) = S(2*n, 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). S(n, 171) = A097844(n).

a(n) = (-2/13)*i*((-1)^n)*T(2*n+1, 13*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

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

a(n) = 171*a(n-1) - a(n-2), n > 1, a(0)=1, a(1)=172. - Philippe Deléham, Nov 18 2008

Example

All positive solutions of Pell equation x^2 - 173*y^2 = -4 are (13 = 13*1,1), (2236 = 13*172,170), (382343 = 13*29411,29069), (65378417 = 13*5029109,4970629), ...

Mathematica

LinearRecurrence[{171, -1}, {1, 172}, 20] (* Harvey P. Dale, Feb 27 2012 *)
CoefficientList[Series[(1+x)/(1-171*x+x^2), {x, 0, 20}], x] (* Stefano Spezia, Jan 14 2019 *)

Prog

(PARI) Vec((1+x)/(1-171*x+x^2)+O(x^20)) \\ Charles R Greathouse IV, Feb 08 2017
(Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/(1-171*x+x^2) )); // G. C. Greubel, Jan 14 2019
(Sage) ((1+x)/(1-171*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 14 2019
(GAP) a:=[1, 172];; for n in [3..20] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 14 2019

Crossrefs

Cf. A049310, A098244.

Sequence in context: A250340 A035828 A259017 * A364937 A261530 A246135

Adjacent sequences: A097842 A097843 A097844 * A097846 A097847 A097848

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 10 2004

Status

approved