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

1, 39, 1481, 56239, 2135601, 81096599, 3079535161, 116941239519, 4440687566561, 168629186289799, 6403468391445801, 243163169688650639, 9233796979777278481, 350641122061847931639, 13315128841370444123801, 505624254850015028772799, 19200406555459200649242561, 729109824852599609642444519, 27686972937843325965763649161

Offset

0,2

Links

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

Christian Aebi and Grant Cairns, Lattice equable quadrilaterals III: tangential and extangential cases, Integers (2023) Vol. 23, #A48.

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

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

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

a(n) = (-1)^n*T(2*n + 1, 3*i)/(3*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

a(n) = ((3 + sqrt(10))*(19 + 6*sqrt(10))^n - ((-3 + sqrt(10))*(19 - 6*sqrt(10))^n))/6. - Gerry Martens, Jul 09 2015

a(n) = (1/3)*sinh((2*n + 1)*arcsinh(3)). - Bruno Berselli, Apr 03 2018

Example

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

Mathematica

LinearRecurrence[{38, -1}, {1, 39}, 20] (* Ray Chandler, Aug 11 2015 *)

Prog

(PARI) Vec((1+x)/(1-38*x+x^2) + O(x^20)) \\ Michel Marcus, Jul 10 2015

Crossrefs

Cf. A078987 for S(n, 38).

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Sequence in context: A078970 A020303 A235973 * A162871 A163222 A163668

Adjacent sequences: A097311 A097312 A097313 * A097315 A097316 A097317

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Extensions

More terms from Indranil Ghosh, Feb 04 2017

Status

approved