A111258 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2.

0, 539, 560, 1803, 4740, 4859, 12020, 29103, 29796, 71519, 171080, 175119, 418296, 998579, 1022120, 2439459, 5821596, 5958803, 14219660, 33932199, 34731900, 82879703, 197772800, 202433799, 483059760, 1152705803, 1179872096

Offset: 1

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+601, y).

Corresponding values y of solutions (x, y) are in A160098.

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

lim_{n -> infinity} a(n)/a(n-1) = (843+418*sqrt(2))/601 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (361299+5950*sqrt(2))/601^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A160098, A001652, A101152, A156035 (decimal expansion of 3+2*sqrt(2)), A160099 (decimal expansion of (843+418*sqrt(2))/601), A160100 (decimal expansion of (361299+5950*sqrt(2))/601^2).

Programs

Magma

I:=[0,539,560,1803,4740,4859,12020]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) -Self(n-6) + Self(n-7): n in [1..30]]; // G. C. Greubel, Apr 22 2018

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,539,560,1803,4740,4859,12020}, 50] (* G. C. Greubel, Apr 22 2018 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1202*n+361201), print1(n, ",")))}

PARI

x='x+O('x^30); concat([0], Vec(x*(539 +21*x +1243*x^2 -297*x^3 -7*x^4 -297*x^5)/((1-x)*(1 -6*x^3 +x^6)))) \\ G. C. Greubel, Apr 22 2018

Formula

a(n) = 6*a(n-3) - a(n-6) + 1202 for n > 6; a(1)=0, a(2)=539, a(3)=560, a(4)=1803, a(5)=4740, a(6)=4859.

G.f.: x*(539 +21*x +1243*x^2 -297*x^3 -7*x^4 -297*x^5)/((1-x)*(1 -6*x^3 +x^6)).

a(3*k+1) = 601*A001652(k) for k >= 0.

Extensions

Edited and one term added by Klaus Brockhaus, May 18 2009