A097315 Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n) = A097314(n), n >= 0.

1, 37, 1405, 53353, 2026009, 76934989, 2921503573, 110940200785, 4212806126257, 159975692596981, 6074863512559021, 230684837784645817, 8759948972303982025, 332647376109766671133, 12631840343198829521029, 479677285665445755127969, 18215105014943739865341793, 691694313282196669127860165

Offset

0,2

Comments

Hypotenuses of primitive Pythagorean triples in A195616 and A195617. - Clark Kimberling, Sep 22 2011

Links

Indranil Ghosh, Table of n, a(n) for n = 0..631

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.

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 (38,-1).

Formula

a(n) = S(n, 38) - S(n-1, 38) = T(2*n+1, sqrt(10))/sqrt(10), with Chebyshev polynomials of the second and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.

a(n) = ((-1)^n)*S(2*n, 6*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

G.f.: (1-x)/(1-38*x+x^2).

a(n) = 38*a(n-1) - a(n-2) for n > 1. - Philippe Deléham, Nov 18 2008

a(n) = sqrt(2+(19-6*sqrt(10))^(1+2*n) + (19+6*sqrt(10))^(1+2*n))/(2*sqrt(10)). - Gerry Martens, Jun 04 2015

a(n) = A078987(n) - A078987(n-1). - R. J. Mathar, Dec 05 2015

a(n) = A005668(2*n+1). - Michael Somos, Feb 24 2023

Example

(x,y) = (3,1), (117,37), (4443,1405), ... give the positive integer solutions to x^2 - 10*y^2 = -1.
G.f. = 1 + 37*x + 1405*x^2 + 53353*x^3 + ... - Michael Somos, Feb 24 2023

Mathematica

CoefficientList[Series[(1-x)/(1-38x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Feb 04 2017 *)
LinearRecurrence[{38, -1}, {1, 37}, 21] (* G. C. Greubel, Aug 01 2019 *)

Prog

(PARI) Vec((1-x)/(1-38*x+x^2) + O(x^20)) \\ Michel Marcus, Jun 04 2015
(Magma) I:=[1, 37]; [n le 2 select I[n] else 38*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-38*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 37];; for n in [3..20] do a[n]:=38*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Crossrefs

Cf. A078987 (partial sums), A097314, A195616, A195617, A049310, A053120, A005668.

Row 3 of array A188647.

Cf. A221874.

Cf. similar sequences listed in A238379.

Sequence in context: A189061 A009981 A370334 * A158741 A094490 A231543

Adjacent sequences: A097312 A097313 A097314 * A097316 A097317 A097318

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Extensions

Typo in recurrence formula corrected by Laurent Bonaventure (bonave(AT)free.fr), Oct 03 2010

More terms added by Indranil Ghosh, Feb 04 2017

Status

approved