A157213 Positive numbers y such that y^2 is of the form x^2+(x+137)^2 with integer x.

97, 137, 277, 305, 685, 1565, 1733, 3973, 9113, 10093, 23153, 53113, 58825, 134945, 309565, 342857, 786517, 1804277, 1998317, 4584157, 10516097, 11647045, 26718425, 61292305, 67883953, 155726393, 357237733, 395656673, 907639933

Offset: 1

Author

Klaus Brockhaus, Feb 25 2009

Keywords

• nonn
• easy

Comments

(-65, a(1)) and (A129544(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

Examples

			(-65, a(1)) = (-65, 97) is a solution: (-65)^2+(-65+137)^2 = 4225+5184 = 9409 = 97^2.
(A129544(1), a(2)) = (0, 137) is a solution: 0^2+(0+137)^2 = 18769 = 137^2.
(A129544(3), a(4)) = (136, 305) is a solution: 136^2+(136+137)^2 = 18496+74529 = 93025 = 305^2.
		

Links

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

Crossrefs

Cf. A129544, A001653, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))).

Programs

PARI

{forstep(n=-68, 1000000000, [3, 1], if(issquare(n^2+(n+137)^2,&k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=97, a(2)=137, a(3)=277, a(4)=305, a(5)=685, a(6)=1565.

G.f.: x*(1-x)*(97+234*x+511*x^2+234*x^3+97*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 137*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) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 1.

Limit_{n -> oo} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}.