A159574 Positive numbers y such that y^2 is of the form x^2+(x+337)^2 with integer x.

313, 337, 365, 1513, 1685, 1877, 8765, 9773, 10897, 51077, 56953, 63505, 297697, 331945, 370133, 1735105, 1934717, 2157293, 10112933, 11276357, 12573625, 58942493, 65723425, 73284457, 343542025, 383064193, 427133117, 2002309657

Offset: 1

Author

Klaus Brockhaus, Apr 16 2009

Keywords

• nonn
• easy

Comments

(-25,a(1)) and (A129999(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

Examples

			(-25, a(1)) = (-25, 313) is a solution: (-25)^2+(-25+337)^2 = 625+97344 = 97969 = 313^2.
(A129993(1), a(2)) = (0, 337) is a solution: 0^2+(0+337)^2 = 113569 = 337^2.
(A129993(3), a(4)) = (888, 1513) is a solution: 888^2+(888+337)^2 = 788544+1500625 = 2289169 = 1513^2.
		

Links

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

Crossrefs

Cf. A129999, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576(decimal expansion of (278307+179662*sqrt(2))/337^2).

Programs

PARI

{forstep(n=-28, 50000000, [3, 1], if(issquare(2*n^2+674*n+113569, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)for n > 6; a(1)=313, a(2)=337, a(3)=365, a(4)=1513, a(5)=1685, a(6)=1877.

G.f.: x*(1-x)*(313+650*x+1015*x^2+650*x^3+313*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 337*A001653(k) for k >= 1.

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

Limit_{n -> oo} a(n)/a(n-1) = (339+26*sqrt(2))/337 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 1.