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

1,1

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.

Links

Table of n, a(n) for n=1..30.

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.

Example

(-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.

Prog

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

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).

Sequence in context: A325074 A182476 A182451 * A283792 A333743 A141948

Adjacent sequences: A161480 A161481 A161482 * A161484 A161485 A161486

Keyword

nonn

Author

Klaus Brockhaus, Jun 13 2009

Status

approved