A253811 Part of the y solutions of the Pell equation x^2 - 2*y^2 = +7.

3, 19, 111, 647, 3771, 21979, 128103, 746639, 4351731, 25363747, 147830751, 861620759, 5021893803, 29269742059, 170596558551, 994309609247, 5795261096931, 33777256972339, 196868280737103, 1147432427450279, 6687726283964571, 38978925276337147, 227185825374058311

Offset: 0

Author

Wolfdieter Lang, Feb 05 2015

Keywords

• nonn
• easy

Comments

All positive solutions y = a(n) of the (generalized) Pell equation x^2 - 2*y^2 = +7 based on the fundamental solution (x2,y2) = (5,3) of the second class of (proper) solutions. The corresponding x solutions are given by x(n) = A101386(n).

All other positive solutions come from the first class of (proper) solutions based on the fundamental solution (x1,y1) = (3,1). These are given in A038762 and A038761.

All solutions of this Pell equation are found in A077443(n+1) and A077442(n), for n >= 0. See the Nagell reference on how to find all solutions.

Examples

			A101386(2)^2 - 2*a(2) = 157^2 - 2*111^2 = +7.
		

References

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-1).

Crossrefs

Cf. A001109, A038761, A038762, A077442, A077443, A101386.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x+3)/(x^2-6*x+1))); // G. C. Greubel, Jul 26 2018

Mathematica

LinearRecurrence[{6,-1}, {3,19}, 30] (* or *) CoefficientList[Series[ (x+3)/(x^2-6*x+1), {z, 0, 50}], x]  (* G. C. Greubel, Jul 26 2018 *)

PARI

Vec((x+3)/(x^2-6*x+1) + O(x^100)) \\ Colin Barker, Feb 05 2015

Formula

a(n) = irrational part of z(n), where z(n) = (5+3*sqrt(2))*(3+2*sqrt(2))^n, n >= 0, the general positive solutions of the second class of proper solutions.

From Colin Barker, Feb 05 2015: (Start)

a(n) = 6*a(n-1) - a(n-2).

G.f.: (x+3) / (x^2-6*x+1). (End)

a(n) = 3*A001109(n+1) + A001109(n). - R. J. Mathar, Feb 05 2015

E.g.f.: exp(3*x)*(6*cosh(2*sqrt(2)*x) + 5*sqrt(2)*sinh(2*sqrt(2)*x))/2. - Stefano Spezia, Mar 16 2024