A254757 Part of the positive proper solutions x of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (-1, 5).

17, 103, 601, 3503, 20417, 118999, 693577, 4042463, 23561201, 137324743, 800387257, 4664998799, 27189605537, 158472634423, 923646201001, 5383404571583, 31376781228497, 182877282799399, 1065886915567897, 6212444210607983

Offset: 1

Author

Wolfdieter Lang, Feb 07 2015

Keywords

• nonn
• easy

Comments

The corresponding y solutions are given in A220414.

The other part of the proper (sometimes called primitive) solutions are given in (A254758(n), A254759(n)) for n >= 1.

The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 0.

Examples

			The first pairs of positive solutions of this part of the Pell equation  x^2 - 2*y^2 = - 7^2 are: [17, 13], [103, 73], [601, 425], [3503, 2477], [20417, 14437], [118999, 84145], [693577, 490433], [4042463, 2858453], [23561201, 16660285], [137324743, 97103257], ...
		

References

• T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

Links

• Wolfdieter Lang, Binary Quadratic Forms (indefinite case).
• Index entries for linear recurrences with constant coefficients, signature (6, -1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A049310, A220414, A254758, A254759, A001653, A002315.

Programs

Mathematica

LinearRecurrence[{6,-1},{17,103},20] (* Harvey P. Dale, Sep 01 2017 *)

PARI

Vec((17 + x)/(1 - 6*x + x^2) + O(x^30)) \\ Michel Marcus, Feb 08 2015

Formula

a(n) = rational part of z(n), where z(n) = (-1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 1.

G.f.: (17 + x)/(1 - 6*x + x^2).

a(n) = 6*a(n-1) - a(n-2), n >= 2, with a(0) = -1 and a(1) = 17.

a(n) = 17*S(n-1, 6) + S(n-2, 6), n >= 1, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).