A097738 Pell equation solutions (9*a(n))^2 - 82*b(n)^2 = -1 with b(n):=A097739(n), n >= 0.

1, 327, 106601, 34751599, 11328914673, 3693191431799, 1203969077851801, 392490226188255327, 127950609768293384801, 41711506294237455189799, 13597823101311642098489673, 4432848619521301086652443599, 1445095052140842842606598123601

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Examples

			(x,y) = (9*1=9;1), (2943=9*327;325), (959409=9*106601;105949), ... give the positive integer solutions to x^2 - 82*y^2 =-1.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..397
• 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 (326,-1).

Crossrefs

Cf. A097737 for S(n, 2*163).

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

Programs

Magma

a:=[1,327]; [n le 2 select a[n] else 326*Self(n-1) - Self(n-2): n in [1..13]]; // Marius A. Burtea, Jan 23 2020

Magma

R:=PowerSeriesRing(Integers(), 13); Coefficients(R!( (1 + x)/(1 - 2*163*x + x^2))); // Marius A. Burtea, Jan 23 2020

Mathematica

LinearRecurrence[{326, -1}, {1, 327}, 12] (* Ray Chandler, Aug 12 2015 *)

PARI

x='x+O('x^99); Vec((1+x)/(1-2*163*x+x^2)) \\ Altug Alkan, Apr 05 2018

Formula

G.f.: (1 + x)/(1 - 2*163*x + x^2).

a(n) = S(n, 2*163) + S(n-1, 2*163) = S(2*n, 2*sqrt(82)), with Chebyshev polynomials of the 2nd 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, 9*i)/(9*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

a(n) = 326*a(n-1) - a(n-2), n>1; a(0)=1, a(1)=327. - Philippe Deléham, Nov 18 2008

a(n) = (1/9)*sinh((2*n + 1)*arcsinh(9)). - Bruno Berselli, Apr 03 2018