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

1,1

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.

Links

Table of n, a(n) for n=1..27.

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.

Example

(-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.

Prog

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

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).

Sequence in context: A020378 A308799 A308791 * A363717 A112371 A031937

Adjacent sequences: A160197 A160198 A160199 * A160201 A160202 A160203

Keyword

nonn

Author

Klaus Brockhaus, May 18 2009

Status

approved