A160574 Positive numbers y such that y^2 is of the form x^2+(x+313)^2 with integer x.

233, 313, 493, 905, 1565, 2725, 5197, 9077, 15857, 30277, 52897, 92417, 176465, 308305, 538645, 1028513, 1796933, 3139453, 5994613, 10473293, 18298073, 34939165, 61042825, 106648985, 203640377, 355783657, 621595837, 1186903097

Offset: 1

Author

Klaus Brockhaus, Jun 08 2009

Keywords

• nonn

Comments

(-105, a(1)) and (A129640(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+313)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (363+130*sqrt(2))/313 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (119187+47998*sqrt(2))/313^2 for n mod 3 = 1.

Examples

			(-105, a(1)) = (-105, 233) is a solution: (-105)^2+(-105+313)^2 = 11025+43264 = 54289 = 233^2.
(A129640(1), a(2)) = (0, 313) is a solution: 0^2+(0+313)^2 = 97969 = 313^2.
(A129640(3), a(4)) = (464, 905) is a solution: 464^2+(464+313)^2 = 215296+603729 = 819025 = 905^2.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A129640, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160575 (decimal expansion of (363+130*sqrt(2))/313), A160576 (decimal expansion of (119187+47998*sqrt(2))/313^2).

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{233,313,493,905,1565,2725},30] (* Harvey P. Dale, Dec 21 2022 *)

PARI

{forstep(n=-108, 10000000, [3, 1], if(issquare(2*n^2+626*n+97969, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=233, a(2)=313, a(3)=493, a(4)=905, a(5)=1565, a(6)=2725.

G.f.: (1-x)*(233+546*x+1039*x^2+546*x^3+233*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 313*A001653(k) for k >= 1.