A159641 Positive numbers y such that y^2 is of the form x^2+(x+647)^2 with integer x.

613, 647, 685, 2993, 3235, 3497, 17345, 18763, 20297, 101077, 109343, 118285, 589117, 637295, 689413, 3433625, 3714427, 4018193, 20012633, 21649267, 23419745, 116642173, 126181175, 136500277, 679840405, 735437783, 795581917

Offset: 1

Author

Klaus Brockhaus, Apr 21 2009

Keywords

• nonn
• easy

Comments

(-35,a(1)) and (A130013(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.

Examples

			(-35, a(1)) = (-35, 613) is a solution: (-35)^2+(-35+647)^2 = 1225+374544 = 375769 = 613^2.
(A130013(1), a(2)) = (0, 647) is a solution: 0^2+(0+647)^2 = 418609 = 647^2.
(A130013(3), a(4)) = (1768, 2993) is a solution: 1768^2+(1768+647)^2 = 3125824+5832225 = 8958049 = 2993^2.
		

Links

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

Crossrefs

Cf. A130013, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{613,647,685,2993,3235,3497},30] (* Harvey P. Dale, Jun 22 2022 *)

PARI

{forstep(n=-36, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=613, a(2)=647, a(3)=685, a(4)=2993, a(5)=3235, a(6)=3497.

G.f.: (1-x)*(613+1260*x+1945*x^2+1260*x^3+613*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 647*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) = (649+36*sqrt(2))/647 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 for n mod 3 = 1.