A160580 Positive numbers y such that y^2 is of the form x^2+(x+457)^2 with integer x.

325, 457, 877, 1073, 2285, 4937, 6113, 13253, 28745, 35605, 77233, 167533, 207517, 450145, 976453, 1209497, 2623637, 5691185, 7049465, 15291677, 33170657, 41087293, 89126425, 193332757, 239474293, 519466873, 1126825885, 1395758465

Offset

1,1

Comments

(-204, a(1)) and (A129642(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 for n mod 3 = 1.

Links

Table of n, a(n) for n=1..28.

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=325, a(2)=457, a(3)=877, a(4)=1073, a(5)=2285, a(6)=4937.

G.f.: (1-x)*(325+782*x+1659*x^2+782*x^3+325*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 457*A001653(k) for k >= 1.

Example

(-204, a(1)) = (-204, 325) is a solution: (-204)^2+(-204+457)^2 = 41616+64009 = 105625 = 325^2.
(A129642(1), a(2)) = (0, 457) is a solution: 0^2+(0+457)^2 = 208849 = 457^2.
(A129642(3), a(4)) = (495, 1073) is a solution: 495^2+(495+457)^2 = 245025+906304 = 1151329 = 1073^2.

Prog

(PARI) {forstep(n=-204, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849, &k), print1(k, ", ")))}

Crossrefs

Cf. A129642, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).

Sequence in context: A025286 A025304 A351801 * A158272 A183645 A299708

Adjacent sequences: A160577 A160578 A160579 * A160581 A160582 A160583

Keyword

nonn

Author

Klaus Brockhaus, Jun 08 2009

Status

approved