A157119 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

0, 84, 105, 309, 765, 884, 2060, 4712, 5405, 12257, 27713, 31752, 71688, 161772, 185313, 418077, 943125, 1080332, 2436980, 5497184, 6296885, 14204009, 32040185, 36701184, 82787280, 186744132, 213910425, 482519877, 1088424813, 1246761572

Offset

1,2

Comments

Corresponding values y of solutions (x, y) are in A157120.

lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

lim_{n -> infinity} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*sqrt(2))^2 for n mod 3 = 0.

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Formula

a(n) = 6*a(n-3)-a(n-6)+206 for n > 6; a(1) = 0, a(2) = 84, a(3) = 105, a(4) = 309, a(5) = 765, a(6) = 884.

G.f.: x*(84+21*x+204*x^2-48*x^3-7*x^4-48*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 103*A001652(k) for k >= 0.

Prog

(PARI) {forstep(n=0, 1300000000, [1, 3], if(issquare(2*n^2+206*n+10609), print1(n, ", ")))}

Crossrefs

Cf. A157120, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

Sequence in context: A352230 A214866 A111313 * A209204 A219801 A316833

Adjacent sequences: A157116 A157117 A157118 * A157120 A157121 A157122

Keyword

nonn,easy

Author

Klaus Brockhaus, Feb 25 2009

Status

approved