A254758 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).

1, 23, 137, 799, 4657, 27143, 158201, 922063, 5374177, 31322999, 182563817, 1064059903, 6201795601, 36146713703, 210678486617, 1227924205999, 7156866749377, 41713276290263, 243122790992201, 1417023469662943

Offset

0,2

Comments

The corresponding y solutions are given in A254759.

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.

References

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

Links

Table of n, a(n) for n=0..19.

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.

Formula

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

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

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

a(n) = S(n, 6) + 17*S(n-1, 6), n >= 0, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).

Example

The first pairs of positive solutions of this part of the Pell equation  x^2 - 2*y^2 = - 7^2 are: [1, 5], [23, 17], [137, 97], [799, 565], [4657, 3293], [27143, 19193], [158201, 111865], [922063, 651997], [5374177, 3800117], ...

Maple

with(orthopoly): a := n -> `if`(n=0, 1, U(n, 3)+17*U(n-1, 3)):
seq(a(n), n=0..19); # Peter Luschny, Feb 07 2015

Mathematica

LinearRecurrence[{6, -1}, {1, 23}, 20] (* Jean-François Alcover, Jun 28 2019 *)

Prog

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

Crossrefs

Cf. A254759, A254757, A220414, A001653, A002315, A049310.

Sequence in context: A260842 A093806 A142679 * A057883 A160221 A042024

Adjacent sequences: A254755 A254756 A254757 * A254759 A254760 A254761

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 07 2015

Status

approved