A118611 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.

0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860, 9177, 10904, 14553, 19404, 25853, 30660, 40817, 54320, 64385, 85652, 113925, 151512, 179529, 238728, 317429, 376092, 500045, 664832, 883905, 1047200, 1392237, 1850940, 2192853

Offset: 1

Author

Mohamed Bouhamida, May 08 2006

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+343, y); 343=7^3.

Corresponding values y of solutions (x, y) are in A157246.

Limit_{n -> oo} a(n)/a(n-7) = 3+2*sqrt(2).

Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 7 = {1, 2, 4, 5, 6}.

Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 7 = {0, 3}.

Examples

			132^2+(132+343)^2 = 17424+225625 = 243049 = 493^2.
		

Links

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

Crossrefs

Cf. A157246, A001652, A118576, A118554, A118611, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

Programs

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, -1, 1}, {0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

PARI

{forstep(n=0, 1400000, [1, 3], if(issquare(n^2+(n+343)^2), print1(n, ",")))}

Formula

a(n) = 6*a(n-7)-a(n-14)+686 for n > 14; a(1)=0, a(2)=77, a(3)=132, a(4)=245, a(5)=392, a(6)=585, a(7)=728, a(8)=1029, a(9)=1428, a(10)=1725, a(11)=2352, a(12)=3185, a(13)=4292, a(14)=5117.

G.f.: x*(77+55*x+113*x^2+147*x^3+193*x^4+143*x^5+301*x^6-63*x^7 -33*x^8-51*x^9-49*x^10-51*x^11-33*x^12-63*x^13)/((1-x)*(1-6*x^7+x^14)).

a(7*k+1) = 343*A001652(k) for k >= 0.

Extensions

Edited by Klaus Brockhaus, Feb 25 2009