A160041 Positive numbers y such that y^2 is of the form x^2+(x+73)^2 with integer x.

53, 73, 125, 193, 365, 697, 1105, 2117, 4057, 6437, 12337, 23645, 37517, 71905, 137813, 218665, 419093, 803233, 1274473, 2442653, 4681585, 7428173, 14236825, 27286277, 43294565, 82978297, 159036077, 252339217, 483632957, 926930185

Offset: 1

Author

Klaus Brockhaus, May 04 2009

Keywords

• nonn

Comments

(-28, a(1)) and (A129289(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 1.

Examples

			(-28, a(1)) = (-28, 53) is a solution: (-28)^2+(-28+73)^2 = 784+2025 = 2809 = 53^2.
(A129289(1), a(2)) = (0, 73) is a solution: 0^2+(0+73)^2 = 5329 = 73^2.
(A129289(3), a(4)) = (95, 193) is a solution: 95^2+(95+73)^2 = 9025+28224 = 37249 = 193^2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A129289, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160042 (decimal expansion of (89+36*sqrt(2))/73), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).

Programs

Magma

I:=[53,73,125,193,365,697]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 21 2018

Mathematica

LinearRecurrence[{0,0,6,0,0,-1}, {53,73,125,193,365,697}, 50] (* G. C. Greubel, Apr 21 2018 *)

PARI

{forstep(n=-28, 10000000, [3, 1], if(issquare(2*n^2+146*n+5329, &k), print1(k, ",")))}

PARI

x='x+O('x^30); Vec((1-x)*(53 +126*x +251*x^2 +126*x^3 +53*x^4)/(1 -6*x^3+x^6)) \\ G. C. Greubel, Apr 21 2018

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=53, a(2)=73, a(3)=125, a(4)=193, a(5)=365, a(6)=697.

G.f.: (1-x)*(53 +126*x +251*x^2 +126*x^3 +53*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 73*A001653(k) for k >= 1.