A097726 Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

1, 103, 10505, 1071407, 109273009, 11144775511, 1136657829113, 115927953794015, 11823514629160417, 1205882564220568519, 122988198035868828521, 12543590317094399940623, 1279323224145592925115025, 130478425272533383961791927, 13307520054574259571177661529

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

a(-1) = -1. - Artur Jasinski, Feb 10 2010

5*a(n) gives the x-values in the solution to the Pell equation x^2 - 26*y^2 = -1. - Colin Barker, Aug 24 2013

Examples

			(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 = -1.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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 (102,-1).

Crossrefs

Cf. A097725 for S(n, 102).

Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121. - Artur Jasinski, Feb 10 2010

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

Programs

Mathematica

Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (* Artur Jasinski, Feb 10 2010 *)
CoefficientList[Series[(1 + x)/(1 - 102 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 13 2014 *)
LinearRecurrence[{102,-1},{1,103},20] (* Harvey P. Dale, Aug 20 2017 *)

PARI

x='x+O('x^99); Vec((1+x)/(1-102*x+x^2)) \\ Altug Alkan, Apr 05 2018

Formula

G.f.: (1 + x)/(1 - 102*x + x^2).

a(n) = S(n, 2*51) + S(n-1, 2*51) = S(2*n, 2*sqrt(26)), 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, 5*i)/(5*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=103. - Philippe Deléham, Nov 18 2008

a(n) = (1/5)*sinh((2*n-1)*arcsinh(5)), n >= 1. - Artur Jasinski, Feb 10 2010

Extensions

More terms from Harvey P. Dale, Aug 20 2017