A129999 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

0, 27, 888, 1011, 1148, 6027, 6740, 7535, 35948, 40103, 44736, 210335, 234552, 261555, 1226736, 1367883, 1525268, 7150755, 7973420, 8890727, 41678468, 46473311, 51819768, 242920727, 270867120, 302028555, 1415846568, 1578730083

Offset: 1

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+337, y).

Corresponding values y of solutions (x, y) are in A159574.

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) = (339+26*sqrt(2))/337 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A159574, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576 (decimal expansion of (278307+179662*sqrt(2))/337^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,27,888,1011,1148,6027,6740},40] (* Harvey P. Dale, Feb 26 2015 *)

PARI

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+674*n+113569), print1(n, ",")))}

Formula

a(n)=6*a(n-3)-a(n-6)+674 for n > 6; a(1)=0, a(2)=27, a(3)=888, a(4)=1011, a(5)=1148, a(6)=6027.

G.f.: x*(27+861*x+123*x^2-25*x^3-287*x^4-25*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 337*A001652(k) for k >= 0.

a(0)=0, a(1)=27, a(2)=888, a(3)=1011, a(4)=1148, a(5)=6027, a(6)=6740, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Feb 26 2015

Extensions

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