A160200 Positive numbers y such that y^2 is of the form x^2+(x+761)^2 with integer x.

541, 761, 1465, 1781, 3805, 8249, 10145, 22069, 48029, 59089, 128609, 279925, 344389, 749585, 1631521, 2007245, 4368901, 9509201, 11699081, 25463821, 55423685, 68187241, 148414025, 323032909, 397424365, 865020329, 1882773769

Offset: 1

Author

Klaus Brockhaus, May 18 2009

Keywords

• nonn

Comments

(-341, a(1)) and (A122694(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+761)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (1003+462*sqrt(2))/761 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (591603+85478*sqrt(2))/761^2 for n mod 3 = 1.

Examples

			(-341, a(1)) = (-341, 541) is a solution: (-341)^2+(-341+761)^2 = 116281+176400 = 292681 = 541^2.
(A122694(1), a(2)) = (0, 761) is a solution: 0^2+(0+761)^2 = 579121 = 761^2.
(A122694(3), a(4)) = (820, 1781) is a solution: 820^2+(820+761)^2 = 672400+2499561 = 3171961 = 1781^2.
		

Crossrefs

Cf. A122694, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160201 (decimal expansion of (1003+462*sqrt(2))/761), A160202 (decimal expansion of (591603+85478*sqrt(2))/761^2).

Programs

PARI

{forstep(n=-344, 10000000, [3, 1], if(issquare(2*n^2+1522*n+579121, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=541, a(2)=761, a(3)=1465, a(4)=1781, a(5)=3805, a(6)=8249.

G.f.: (1-x)*(541+1302*x+2767*x^2+1302*x^3+541*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 761*A001653(k) for k >= 1.