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

1,1

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.

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).

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.

Example

(-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.

Mathematica

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

Prog

(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
(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

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).

Sequence in context: A136073 A133187 A057667 * A013538 A107309 A039389

Adjacent sequences: A160038 A160039 A160040 * A160042 A160043 A160044

Keyword

nonn

Author

Klaus Brockhaus, May 04 2009

Status

approved