A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x.

181, 199, 221, 865, 995, 1145, 5009, 5771, 6649, 29189, 33631, 38749, 170125, 196015, 225845, 991561, 1142459, 1316321, 5779241, 6658739, 7672081, 33683885, 38809975, 44716165, 196324069, 226201111, 260624909, 1144260529, 1318396691

Offset: 1

Author

Klaus Brockhaus, Apr 14 2009

Keywords

• nonn
• easy

Comments

(-19,a(1)) and (A129993(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+199)^2 = y^2.

Examples

			(-19, a(1)) = (-19, 181) is a solution: (-19)^2+(-19+199)^2 = 361+32400 = 32761 = 181^2.
(A129993(1), a(2)) = (0, 199) is a solution: 0^2+(0+199)^2 = 39601 = 199^2.
(A129993(3), a(4)) = (504, 865) is a solution: 504^2+(504+199)^2 = 254016+494209 = 748225 = 865^2.
		

Links

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

Crossrefs

Cf. A129993, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{181,199,221,865,995,1145},30] (* Harvey P. Dale, Aug 09 2025 *)

PARI

{forstep(n=-20, 50000000, [1, 3], if(issquare(2*n^2+398*n+39601, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=181, a(2)=199, a(3)=221, a(4)=865, a(5)=995, a(6)=1145.

G.f.: x*(1-x)*(181+380*x+601*x^2+380*x^3+181*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 199*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) = (201+20*sqrt(2))/199 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 1.