A227111 Nonnegative solutions of the Pell equation x^2 - 89*y^2 = +1. Solutions y = 53000*a(n).

0, 1, 1000002, 1000004000003, 1000006000010000004, 1000008000021000020000005, 1000010000036000056000035000006, 1000012000055000120000126000056000007, 1000014000078000220000330000252000084000008, 1000016000105000364000715000792000462000120000009

Offset

0,3

Comments

The Pell equation x^2 - 89*y^2 = +1 has only proper solutions, namely x(n) = A227110(n) and y(n) = 2^3*5^3*53*a(n), n>= 0, where 2^3*5^3*53 = 53000 is the fundamental positive y solution.

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, p. 92-95.

Links

Table of n, a(n) for n=0..9.

Index entries for linear recurrences with constant coefficients, signature (1000002, -1).

Formula

a(n) = S(n-1, 2*500001) with the Chebyshev S- polynomial (see A049310). S(n, -1) = 0, where 500001 = 3*166667 is the corresponding fundamental x-solution.

a(n) = 500001*a(n-1) - a(n-2), n >= 1 with inputs a(-1) = -1 and a(0) = 0.

O.g.f.: x/(1 - 2*500001*x + x^2).

Mathematica

LinearRecurrence[{1000002, -1}, {0, 1}, 10] (* Ray Chandler, Aug 11 2015 *)

Crossrefs

Cf. A227110 (x solutions), A049310, A227137, A227138.

Sequence in context: A017514 A017646 A227138 * A227137 A096212 A065325

Adjacent sequences: A227108 A227109 A227110 * A227112 A227113 A227114

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 02 2013

Status

approved