A161487 Positive numbers y such that y^2 is of the form x^2+(x+191)^2 with integer x.

149, 191, 269, 625, 955, 1465, 3601, 5539, 8521, 20981, 32279, 49661, 122285, 188135, 289445, 712729, 1096531, 1687009, 4154089, 6391051, 9832609, 24211805, 37249775, 57308645, 141116741, 217107599, 334019261, 822488641, 1265395819

Offset: 1

Author

Klaus Brockhaus, Jun 13 2009

Keywords

• nonn

Comments

(-51, a(1)) and (A161486(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (209+60*sqrt(2))/191 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (52323+26522*sqrt(2))/191^2 for n mod 3 = 1.

Examples

			(-51, a(1)) = (-51, 149) is a solution: (-51)^2+(-51+191)^2 = 2601+19600 = 22201 = 149^2.
(A161486(1), a(2)) = (0, 191) is a solution: 0^2+(0+191)^2 = 36481 = 191^2.
(A161486(3), a(4)) = (336, 625) is a solution: 336^2+(336+191)^2 = 112896+277729 = 390625 = 625^2.
		

Crossrefs

Cf. A161486, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A161488 (decimal expansion of (209+60*sqrt(2))/191), A161489 (decimal expansion of (52323+26522*sqrt(2))/191^2).

Programs

PARI

{forstep(n=-52, 100000000, [1, 3], if(issquare(2*n^2+382*n+36481, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=149, a(2)=191, a(3)=269, a(4)=625, a(5)=955, a(6)=1465.

G.f.: (1-x)*(149+340*x+609*x^2+340*x^3+149*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 191*A001653(k) for k >= 1.