A157246 Positive numbers y such that y^2 is of the form x^2+(x+343)^2 with integer x.

245, 265, 287, 343, 427, 493, 637, 833, 1097, 1295, 1715, 2275, 2693, 3577, 4753, 6317, 7483, 9947, 13223, 15665, 20825, 27685, 36805, 43603, 57967, 77063, 91297, 121373, 161357, 214513, 254135, 337855, 449155, 532117, 707413, 940457, 1250273

Offset: 1

Author

Klaus Brockhaus, Feb 25 2009

Keywords

• nonn

Comments

(-147, a(1)), (-96, a(2)), (-63, a(3)) and (A118611(n), a(n+3)) are solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.

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

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

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

Examples

			(-147, a(1)) = (-147, 245) is a solution: (-147)^2+(-147+343)^2 = 21609+38416 = 60025 = 245^2.
(A118611(1), a(4)) = (0, 343) is a solution: 0^2+(0+343)^2 = 117649 = 343^2.
(A118611(3), a(6)) = (132, 493) is a solution: 132^2+(132+343)^2 = 17424+225625 = 243049 = 493^2.
		

Links

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

Crossrefs

Cf. A118611, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

Programs

Mathematica

Sqrt[#]&/@Select[Table[x^2+(x+343)^2,{x,-150,885000}], IntegerQ[ Sqrt[#]]&] (* or *) LinearRecurrence[{0,0,0,0,0,0,6,0,0,0,0,0,0,-1},{245,265,287,343,427,493,637,833,1097,1295,1715,2275,2693,3577},40](* Harvey P. Dale, Dec 30 2011 *)

PARI

{forstep(n=-148, 1000000, [1, 3], if(issquare(n^2+(n+343)^2, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-7)-a(n-14) for n > 14; a(1)=245, a(2)=265, a(3)=287, a(4)=343, a(5)=427, a(6)=493, a(7)=637, a(8)=833, a(9)=1097, a(10)=1295, a(11)=1715, a(12)=2275, a(13)=2693, a(14)=3577.

G.f.: x * (1-x) * (245 +510*x +797*x^2 +1140*x^3 +1567*x^4 +2060*x^5 +2697*x^6 +2060*x^7 +1567*x^8 +1140*x^9 +797*x^10 +510*x^11 +245*x^12) / (1 -6*x^7 +x^14).

a(7*k-3) = 343*A001653(k) for k >= 1.

Extensions

G.f. adapted to the offset by Bruno Berselli, Apr 01 2011