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

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

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

Examples

			(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
		

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).

Crossrefs

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

Row 3 of array A188647.

Cf. A221874.

Cf. similar sequences listed in A238379.

Programs

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

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

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 *)

PARI

Vec((1-x)/(1-38*x+x^2) + O(x^20)) \\ Michel Marcus, Jun 04 2015

Python

from itertools import islice
def A097315_gen(): # generator of terms
    x, y = 30, 10
    while True:
        yield y//10
        x, y = x*19+y*60, x*6+y*19
A097315_list = list(islice(A097315_gen(),20)) # Chai Wah Wu, Apr 24 2025

Sage

((1-x)/(1-38*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

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

E.g.f.: exp(19*x)*(10*cosh(6*sqrt(10)*x) + 3*sqrt(10)*sinh(6*sqrt(10)*x))/10. - Stefano Spezia, Apr 24 2025

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