A254759 Part of the positive proper solutions y of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (1, 5).

5, 17, 97, 565, 3293, 19193, 111865, 651997, 3800117, 22148705, 129092113, 752403973, 4385331725, 25559586377, 148972186537, 868273532845, 5060669010533, 29495740530353, 171913774171585, 1001986904499157

Offset: 0

Author

Wolfdieter Lang, Feb 07 2015

Keywords

• nonn
• easy

Comments

The corresponding x solutions are given in A254758.

The other part of the proper (sometimes called primitive) solutions are given in (A254757(n), A220414(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 >= 1.

Examples

			A254758(3)^2 - 2*a(3)^2 = 799^2 - 2*565^2 = -49.
See also A254758 for the first pairs of solutions.
		

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. A254758, A254757, A220414, A001653, A002315, A049310.

Programs

PARI

Vec((5-13*x)/(1-6*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 08 2015

Formula

a(n) = irrational part of z(n), where z(n) = (1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 0.

G.f.: (5-13*x)/(1-6*x+x^2).

a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = 13 and a(0) = 5.

a(n) = 5*S(n, 6) - 13*S(n-1, 6), n >= 0, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).