A160583 Positive numbers y such that y^2 is of the form x^2+(x+521)^2 with integer x.

445, 521, 629, 2041, 2605, 3329, 11801, 15109, 19345, 68765, 88049, 112741, 400789, 513185, 657101, 2335969, 2991061, 3829865, 13615025, 17433181, 22322089, 79354181, 101608025, 130102669, 462510061, 592214969, 758293925, 2695706185, 3451681789, 4419660881

Offset: 1

Author

Klaus Brockhaus, Jun 08 2009

Keywords

• nonn

Comments

(-84, a(1)) and (A129725(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (537+92*sqrt(2))/521 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (520659+314170*sqrt(2))/521^2 for n mod 3 = 1.

Examples

			(-84, a(1)) = (-84, 445) is a solution: (-84)^2+(-84+521)^2 = 7056+190969 = 198025 = 445^2.
(A129725(1), a(2)) = (0, 521) is a solution: 0^2+(0+521)^2 = 271441 = 521^2.
(A129725(3), a(4)) = (1159, 2041) is a solution: 1159^2+(1159+521)^2 = 1343281+2822400 = 4165681 = 2041^2.
		

Links

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

Crossrefs

Cf. A129725, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160584 (decimal expansion of (537+92*sqrt(2))/521), A160585 (decimal expansion of (520659+314170*sqrt(2))/521^2).

Programs

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {445, 521, 629, 2041, 2605, 3329}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *)

PARI

{forstep(n=-84, 10000000, [3, 1], if(issquare(2*n^2+1042*n+271441, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=445, a(2)=521, a(3)=629, a(4)=2041, a(5)=2605, a(6)=3329.

G.f.: (1-x)*(445+966*x+1595*x^2+966*x^3+445*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 521*A001653(k) for k >= 1.