A129991 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

0, 23, 620, 723, 840, 4223, 4820, 5499, 25200, 28679, 32636, 147459, 167736, 190799, 860036, 978219, 1112640, 5013239, 5702060, 6485523, 29219880, 33234623, 37800980, 170306523, 193706160, 220320839, 992619740, 1129002819, 1284124536, 5785412399, 6580311236

Offset: 1

Author

Mohamed Bouhamida, Jun 14 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+241, y).

Corresponding values y of solutions (x, y) are in A159565.

For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.

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

lim_{n -> infinity} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 0.

Links

• Vincenzo Librandi, 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. A159565, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).

Programs

Mathematica

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 23, 620, 723, 840, 4223, 4820}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)

PARI

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+482*n+58081), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+482 for n > 6; a(1)=0, a(2)=23, a(3)=620, a(4)=723, a(5)=840, a(6)=4223.

G.f.: x*(23+597*x+103*x^2-21*x^3-199*x^4-21*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 241*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 16 2009