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

1,1

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.

Links

Table of n, a(n) for n=1..29.

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.

Example

(-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.

Prog

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

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).

Sequence in context: A307472 A209619 A031929 * A121947 A141946 A128390

Adjacent sequences: A161484 A161485 A161486 * A161488 A161489 A161490

Keyword

nonn

Author

Klaus Brockhaus, Jun 13 2009

Status

approved