A161483 Positive numbers y such that y^2 is of the form x^2+(x+151)^2 with integer x.

109, 151, 265, 389, 755, 1481, 2225, 4379, 8621, 12961, 25519, 50245, 75541, 148735, 292849, 440285, 866891, 1706849, 2566169, 5052611, 9948245, 14956729, 29448775, 57982621, 87174205, 171640039, 337947481, 508088501, 1000391459, 1969702265

Offset: 1

Author

Klaus Brockhaus, Jun 13 2009

Keywords

• nonn

Comments

(-60, a(1)) and (A161482(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+151)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (187+78*sqrt(2))/151 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (24723+6758*sqrt(2))/151^2 for n mod 3 = 1.

Examples

			(-60, a(1)) = (-60, 109) is a solution: (-60)^2+(-60+151)^2 = 3600+8281 = 11881 = 109^2.
(A161482(1), a(2)) = (0, 151) is a solution: 0^2+(0+151)^2 = 22801 = 151^2.
(A161482(3), a(4)) = (189, 389) is a solution: 189^2+(189+151)^2 = 35721+115600 = 151321 = 389^2.
		

Crossrefs

Cf. A161482, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A161484 (decimal expansion of (187+78*sqrt(2))/151), A161485 (decimal expansion of (24723+6758*sqrt(2))/151^2).

Programs

PARI

{forstep(n=-60, 100000000, [1, 3], if(issquare(2*n^2+302*n+22801, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=109, a(2)=151, a(3)=265, a(4)=389, a(5)=755, a(6)=1481.

G.f.: (1-x)*(109+260*x+525*x^2+260*x^3+109*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 151*A001653(k) for k >= 1.