A118675 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.

0, 16, 85, 141, 225, 616, 940, 1428, 3705, 5593, 8437, 21708, 32712, 49288, 126637, 190773, 287385, 738208, 1112020, 1675116, 4302705, 6481441, 9763405, 25078116, 37776720, 56905408, 146166085, 220178973, 331669137, 851918488, 1283297212, 1933109508

Offset: 1

Author

Mohamed Bouhamida, May 19 2006

Keywords

• nonn

Comments

Also values x of Pythagorean triples (x, x+47, y).

Corresponding values y of solutions (x, y) are in A159750.

For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.

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

lim_{n -> infinity} a(n)/a(n-1) = (51+14*sqrt(2))/47 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A159750, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3 +x^6)))); // G. C. Greubel, May 07 2018

Mathematica

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+47)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,16,85,141,225,616,940},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

PARI

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+94+2209), print1(n, ",")))}

Formula

a(n) = 6*a(n-3) -a(n-6) +94 for n > 6; a(1)=0, a(2)=16, a(3)=85, a(4)=141, a(5)=225, a(6)=616.

G.f.: x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 47*A001652(k) for k >= 0.

Extensions

Edited by Klaus Brockhaus, Apr 30 2009