A159565 Positive numbers y such that y^2 is of the form x^2+(x+241)^2 with integer x.

221, 241, 265, 1061, 1205, 1369, 6145, 6989, 7949, 35809, 40729, 46325, 208709, 237385, 270001, 1216445, 1383581, 1573681, 7089961, 8064101, 9172085, 41323321, 47001025, 53458829, 240849965, 273942049, 311580889, 1403776469, 1596651269

Offset: 1

Author

Klaus Brockhaus, Apr 16 2009

Keywords

• nonn
• easy

Comments

(-21,a(1)) and (A129991(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

Examples

			(-21, a(1)) = (-21, 221) is a solution: (-21)^2+(-21+241)^2 = 441+48400 = 48841 = 221^2.
(A129993(1), a(2)) = (0, 241) is a solution: 0^2+(0+241)^2 = 58081= 241^2.
(A129993(3), a(4)) = (620, 1061) is a solution: 620^2+(620+241)^2 = 384400+741321 = 1125721 = 1061^2.
		

Links

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

Crossrefs

Cf. A129991, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{221,241,265,1061,1205,1369},30] (* Harvey P. Dale, Nov 21 2011 *)

PARI

{forstep(n=-24, 50000000, [3, 1], if(issquare(2*n^2+482*n+58081, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=221, a(2)=241, a(3)=265, a(4)=1061, a(5)=1205, a(6)=1369.

G.f.: x*(1-x)*(221+462*x+727*x^2+462*x^3+221*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 241*A001653(k) for k >= 1.

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

Limit_{n -> oo} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 1.