A129992 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.

0, 17, 308, 381, 468, 2117, 2540, 3045, 12648, 15113, 18056, 74025, 88392, 105545, 431756, 515493, 615468, 2516765, 3004820, 3587517, 14669088, 17513681, 20909888, 85498017, 102077520, 121872065, 498319268, 594951693, 710322756, 2904417845, 3467632892

Offset: 1

Author

Mohamed Bouhamida, Jun 14 2007

Keywords

• nonn

Comments

Also values x of Pythagorean triples (x, x+127, y).

Corresponding values y of solutions (x, y) are in A159466.

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) = (129+16*sqrt(2))/127 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (34947+21922*sqrt(2))/127^2 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. A159466, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159467 (decimal expansion of (129+16*sqrt(2))/127), A159468 (decimal expansion of (34947+21922*sqrt(2))/127^2).

Programs

Magma

I:=[0,17,308,381,468,2117,2540]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,17,308,381,468,2117,2540},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

PARI

{forstep(n=0, 500000000, [1, 3], if(issquare(2*n^2+254*n+16129), print1(n, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) + 254 for n > 6; a(1)=0, a(2)=17, a(3)=308, a(4)=381, a(5)=468, a(6)=2117.

G.f.: x*(17+291*x+73*x^2-15*x^3-97*x^4-15*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 127*A001652(k) for k >= 0.

Extensions

Edited and two more terms added by Klaus Brockhaus, Apr 13 2009