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A097726 Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n>=0. 4
1, 103, 10505, 1071407, 109273009, 11144775511, 1136657829113, 115927953794015, 11823514629160417, 1205882564220568519, 122988198035868828521, 12543590317094399940623, 1279323224145592925115025, 130478425272533383961791927, 13307520054574259571177661529 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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]

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).

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

EXAMPLE

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

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 *)

PROG

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

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.

Sequence in context: A034180 A076460 A245495 * A262273 A088584 A238490

Adjacent sequences:  A097723 A097724 A097725 * A097727 A097728 A097729

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

EXTENSIONS

More terms from Harvey P. Dale, Aug 20 2017

STATUS

approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)