A227150 Nonnegative solutions of the Pell equation x^2 - 97*y^2 = +1. Solutions x = a(n).

1, 62809633, 7890099995189377, 991148570062293006927649, 124507355868174813917084187296257, 15640522655760911892802130406069321959713, 1964750975873056298957118436671896252612641334401, 246810575461957025811147127728879197279220180679392209953

Offset

0,2

Comments

The Pell equation x^2 - 97*y^2 = +1 has only proper solutions, namely x(n) = a(n) and y(n) = 6377352*A227151(n), n>= 0.

References

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. VI, 56., pp. 115-200.

O. Perron, Die Lehre von den Kettenbruechen, Band I, Teubner, Stuttgart, 1954, Paragraph 27, pp. 92-95.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (125619266,-1).

Formula

a(n) = (S(n, 2*62809633) - S(n-2, 2*62809633))/2 = T(n,62809633) with the Chebyshev S- and T-polynomials (see A049310 and A053120, respectively). S(n, -2) = -1, S(n, -1)

a(n) = 2*62809633*a(n-1) - a(n-2), n >= 1, with input a(-1) = 62809633 and a(0) = 1. 0.

O.g.f.: (1 - 62809633*x)/(1 - 2*62809633*x + x^2).

a(n+2) = 2*a(1)*a(n+1) + a(n) for n >= 0. - A.H.M. Smeets, Dec 26 2017

Example

n=0: 1^2 - 97*0^2  = +1 (a proper, but not a positive solution),
n=1: 62809633^2 - 97*6377352^2 = +1, where 62809633 is prime and 6377352 = 2^3*3*467*569  is the positive fundamental y-solution.
n=2: 7890099995189377^2 - 97*801118277263632^2 = +1, where  801118277263632 = 6377352*125619266 = (2^3*3*467*569)*(2*62809633).

Mathematica

CoefficientList[Series[(1 - 62809633 x) / (1 - 62809633  2 x + x^2), {x, 0, 10}], x] (* Vincenzo Librandi, Jul 11 2013 *)
LinearRecurrence[{125619266, -1}, {1, 62809633}, 20] (* Harvey P. Dale, Nov 27 2015 *)

Crossrefs

Cf. A227151 (y/6377352 solutions), A049310, A053120, A227274, A227275.

Sequence in context: A205369 A157855 A017539 * A186140 A172575 A172583

Adjacent sequences: A227147 A227148 A227149 * A227151 A227152 A227153

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 05 2013

Status

approved