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

1,1

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}.

Links

Table of n, a(n) for n=1..37.

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

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.

Example

(-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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A013684 A233821 A216233 * A186460 A171994 A129210

Adjacent sequences: A157243 A157244 A157245 * A157247 A157248 A157249

Keyword

nonn

Author

Klaus Brockhaus, Feb 25 2009

Extensions

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

Status

approved