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A255236 All positive solutions x of the second class of the Pell equation x^2 - 2*y^2 = -7. 7
5, 31, 181, 1055, 6149, 35839, 208885, 1217471, 7095941, 41358175, 241053109, 1404960479, 8188709765, 47727298111, 278175078901, 1621323175295, 9449763972869, 55077260661919, 321013799998645, 1871005539329951, 10905019435981061, 63559111076556415 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

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).

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

(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

CROSSREFS

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

Sequence in context: A239334 A180635 A078526 * A202753 A057426 A329014

Adjacent sequences:  A255233 A255234 A255235 * A255237 A255238 A255239

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Feb 26 2015

STATUS

approved

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Last modified February 26 11:49 EST 2020. Contains 332279 sequences. (Running on oeis4.)