A160209 Positive numbers y such that y^2 is of the form x^2+(x+937)^2 with integer x.

673, 937, 1685, 2353, 4685, 9437, 13445, 27173, 54937, 78317, 158353, 320185, 456457, 922945, 1866173, 2660425, 5379317, 10876853, 15506093, 31352957, 63394945, 90376133, 182738425, 369492817, 526750705, 1065077593, 2153561957

Offset: 1

Author

Klaus Brockhaus, May 18 2009

Keywords

• nonn

Comments

(-385, a(1)) and (A129974(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+937)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (1179+506*sqrt(2))/937 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (933747+224782*sqrt(2))/937^2 for n mod 3 = 1.

Examples

			(-385, a(1)) = (-385, 673) is a solution: (-385)^2+(-385+937)^2 = 148225+304704 = 452929 = 673^2.
(A129974(1), a(2)) = (0, 937) is a solution: 0^2+(0+937)^2 = 877969 = 937^2.
(A129974(3), a(4)) = (1128, 2353) is a solution: 1128^2+(1128+937)^2 = 1272384+4264225 = 5536609 = 2353^2.
		

Links

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

Crossrefs

Cf. A129974, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160210 (decimal expansion of (1179+506*sqrt(2))/937), A160211 (decimal expansion of (933747+224782*sqrt(2))/937^2).

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{673,937,1685,2353,4685,9437},30] (* Harvey P. Dale, Dec 25 2017 *)

PARI

{forstep(n=-388, 10000000, [3, 1], if(issquare(2*n^2+1874*n+877969, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=673, a(2)=937, a(3)=1685, a(4)=2353, a(5)=4685, a(6)=9437.

G.f.: (1-x)*(673+1610*x+3295*x^2+1610*x^3+673*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 937*A001653(k) for k >= 1.