A160580 Positive numbers y such that y^2 is of the form x^2+(x+457)^2 with integer x.

325, 457, 877, 1073, 2285, 4937, 6113, 13253, 28745, 35605, 77233, 167533, 207517, 450145, 976453, 1209497, 2623637, 5691185, 7049465, 15291677, 33170657, 41087293, 89126425, 193332757, 239474293, 519466873, 1126825885, 1395758465

Offset: 1

Author

Klaus Brockhaus, Jun 08 2009

Keywords

• nonn

Comments

(-204, a(1)) and (A129642(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 for n mod 3 = 1.

Examples

			(-204, a(1)) = (-204, 325) is a solution: (-204)^2+(-204+457)^2 = 41616+64009 = 105625 = 325^2.
(A129642(1), a(2)) = (0, 457) is a solution: 0^2+(0+457)^2 = 208849 = 457^2.
(A129642(3), a(4)) = (495, 1073) is a solution: 495^2+(495+457)^2 = 245025+906304 = 1151329 = 1073^2.
		

Crossrefs

Cf. A129642, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).

Programs

PARI

{forstep(n=-204, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=325, a(2)=457, a(3)=877, a(4)=1073, a(5)=2285, a(6)=4937.

G.f.: (1-x)*(325+782*x+1659*x^2+782*x^3+325*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 457*A001653(k) for k >= 1.