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A097315 Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n):=A097314(n), n>=0. 19
1, 37, 1405, 53353, 2026009, 76934989, 2921503573, 110940200785, 4212806126257, 159975692596981, 6074863512559021, 230684837784645817, 8759948972303982025, 332647376109766671133, 12631840343198829521029, 479677285665445755127969, 18215105014943739865341793, 691694313282196669127860165 (list; graph; refs; listen; history; text; internal format)
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

Tanya Khovanova, Recursive Sequences

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

EXAMPLE

(x,y) = (3,1), (117,37), (4443,1405),... give the positive integer solutions to x^2 - 10*y^2 =-1.

MATHEMATICA

CoefficientList[Series[(1 - x)/(1 - 38 x + x^2), {x, 0, 17}], x] (* Michael De Vlieger, Feb 04 2017 *)

PROG

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

CROSSREFS

Row 3 of array A188647.

Cf. A221874.

Cf. similar sequences listed in A238379.

Sequence in context: A207185 A189061 A009981 * 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

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Last modified June 24 16:32 EDT 2017. Contains 288707 sequences.