A202637 x-values in the solution to x^2 - 7*y^2 = -3.

2, 5, 37, 82, 590, 1307, 9403, 20830, 149858, 331973, 2388325, 5290738, 38063342, 84319835, 606625147, 1343826622, 9667939010, 21416906117, 154080399013, 341326671250, 2455618445198, 5439809833883, 39135814724155, 86695630670878

Offset

1,1

Comments

The corresponding values of y of this Pell equation are in A202638.

Links

Bruno Berselli, Table of n, a(n) for n = 1..1000

R. A. Mollin, Class Numbers of Quadratic Fields Determinet by Solvability of Diophantine Equations, Mathematics of Computation Vol. 48, 1987, p. 235 (Theorem 1.1, particular case).

Index entries for linear recurrences with constant coefficients, signature (0,16,0,-1).

Formula

G.f.: x*(1+x)*(2+3*x+2*x^2)/(1-16*x^2+x^4).

a(n) = -a(-n+1) = ((-2*(-1)^n+sqrt(7))*(8+3*sqrt(7))^floor(n/2)-(2*(-1)^n+sqrt(7))*(8-3*sqrt(7))^floor(n/2))/2.

a(2n)-a(2n-1) = A202638(2n)+A202638(2n-1).

Mathematica

LinearRecurrence[{0, 16, 0, -1}, {2, 5, 37, 82}, 24]

Prog

(PARI) a=vector(24); a[1]=2; a[2]=5; a[3]=37; a[4]=82; for(i=5, #a, a[i]=16*a[i-2]-a[i-4]); a
(Magma) m:=24; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(2+3*x+2*x^2)/(1-16*x^2+x^4)));
(Maxima) makelist(expand(((-2*(-1)^n+sqrt(7))*(8+3*sqrt(7))^floor(n/2)-(2*(-1)^n+sqrt(7))*(8-3*sqrt(7))^floor(n/2))/2), n, 1, 24);

Crossrefs

Sequence in context: A106129 A163499 A086218 * A138658 A067464 A081545

Adjacent sequences: A202634 A202635 A202636 * A202638 A202639 A202640

Keyword

nonn,easy

Author

Bruno Berselli, Dec 22 2011

Status

approved