A129289 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.

0, 44, 95, 219, 455, 744, 1460, 2832, 4515, 8687, 16683, 26492, 50808, 97412, 154583, 296307, 567935, 901152, 1727180, 3310344, 5252475, 10066919, 19294275, 30613844, 58674480, 112455452, 178430735, 341980107, 655438583, 1039970712, 1993206308, 3820176192

Offset: 1

Author

Mohamed Bouhamida, May 26 2007

Keywords

• nonn

Comments

Also values x of Pythagorean triples (x, x+73, y).

Corresponding values y of solutions (x, y) are in A160041.

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 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A160041, A129288, A001652, 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

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)))); // G. C. Greubel, May 07 2018

Mathematica

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+73)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,44,95,219,455,744,1460},70] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

PARI

{forstep(n=0, 100000000, [3 ,1], if(issquare(2*n^2+146*n+5329), print1(n, ",")))}

Formula

a(n) = 6*a(n-3) -a(n-6) +146 for n > 6; a(1)=0, a(2)=44, a(3)=95, a(4)=219, a(5)=455, a(6)=744.

G.f.: x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 73*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, May 04 2009