A255236 All positive solutions x of the second class of the Pell equation x^2 - 2*y^2 = -7.

5, 31, 181, 1055, 6149, 35839, 208885, 1217471, 7095941, 41358175, 241053109, 1404960479, 8188709765, 47727298111, 278175078901, 1621323175295, 9449763972869, 55077260661919, 321013799998645, 1871005539329951, 10905019435981061, 63559111076556415

Offset: 0

Author

Wolfdieter Lang, Feb 26 2015

Keywords

• nonn
• easy

Comments

For the corresponding y = y2 terms see 2*A038725(n+1).

The Pell equation x^2 - 2*y^2 = 7 has two classes of solutions. See, e.g., the Nagell reference and comments under A254938 and A255233. Here the positive solutions based on the fundamental solution (5, 4) (the second largest positive solution) are considered.

The positive solutions of the first class are given in (A054490(n), 2*A038723(n)), n >= 0.

The combined solutions of both classes are given in (A077446, 4*A077447).

The solutions (x(n), y(n)) of x^2 - 2*y^2 = -7 translate to the solutions (X(n), Y(n)) = (2*y(n) , x(n)) of the Pell equation X^2 - 2*Y^2 = 14.

Examples

			n = 2: 181^2 - 2*(2*64)^2  = -7; (4*64)^2 - 2*181^2 = 14.
n = 2: 2*53 + 75 = 181. - _Wolfdieter Lang_, Mar 19 2015
		

Links

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

Crossrefs

Cf. A038725, A054490, A038723, A077446, 4*A077447, A254938, A255233.

Programs

Magma

I:=[5,31]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2015

Mathematica

CoefficientList[Series[(5 + x) / (1 - 6 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 20 2015 *)

PARI

Vec((5 + x)/(1 - 6*x + x^2) + O(x^30)) \\ Michel Marcus, Mar 20 2015

Formula

a(n) = 5*S(n, 6) + S(n-1, 6), n >= 0, with the Chebyshev polynomials S(n, x) (A049310), with S(-1, x) = 0, evaluated at x = 6. S(n, 6) = A001109(n-1).

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

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

a(n) = 2*A038761(n) + A038762(n), n >= 0. See the Mar 19 comment on A054490. - Wolfdieter Lang, Mar 19 2015

a(n) = ((3-2*sqrt(2))^n*(-8+5*sqrt(2)) + (3+2*sqrt(2))^n*(8+5*sqrt(2))) / (2*sqrt(2)). - Colin Barker, Oct 13 2015