A157247 Positive numbers y such that y^2 is of the form x^2+(x+2401)^2 with integer x.

1715, 1781, 1855, 2009, 2401, 2989, 3451, 3821, 4459, 5831, 6865, 7679, 9065, 12005, 15925, 18851, 21145, 25039, 33271, 39409, 44219, 52381, 69629, 92561, 109655, 123049, 145775, 193795, 229589, 257635, 305221, 405769, 539441, 639079, 717149

Offset: 1

Author

Klaus Brockhaus, Feb 25 2009

Keywords

• nonn

Comments

(-1029, a(1)), (-820, a(2)), (-672, a(3)), (-441, a(3)) and (A118630(n), a(n+4)) are solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.

lim_{n -> infinity} a(n)/a(n-9) = 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 9 = {1, 5, 6}.

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

lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 9 = {3, 8}.

Examples

			(-1029, a(1)) = (-1029, 1715) is a solution: (-1029)^2+(-1029+2401)^2 = 1058841+1882384 = 2941225 = 1715^2.
(A118630(1), a(5)) = (0, 2401) is a solution: 0^2+(0+2401)^2 = 5764801 = 2401^2.
(A118630(3), a(7)) = (924, 3451) is a solution: 924^2+(924+2401)^2 = 853776+11055625 = 11909401 = 3451^2.
		

Crossrefs

Cf. A118630, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

Programs

Mathematica

Sqrt[#]&/@Select[Table[2x^2+4802x+5764801,{x,-1200,510000}], IntegerQ[ Sqrt[ #]]&] (* Harvey P. Dale, Jul 21 2011 *)

PARI

{forstep(n=-1032, 540000, [3 ,1], if(issquare(n^2+(n+2401)^2, &k), print1(k, ",")))}

Formula

a(n)=6*a(n-9)-a(n-18) for n > 18; a(1)=1715, a(2)=1781, a(3)=1855, a(4)=2009, a(5)=2401, a(6)=2989, a(7)=3451, a(8)=3821, a(9)=4459, a(10)=5831, a(11)=6865, a(12)=7679, a(13)=9065, a(14)=12005, a(15)=15925, a(16)=18851, a(17)=21145, a(18)=25039.

G.f.: x * (1-x) * (1715 +3496*x +5351*x^2 +7360*x^3 +9761*x^4 +12750*x^5 +16201*x^6 +20022*x^7 +24481*x^8 +20022*x^9 +16201*x^10 +12750*x^11 +9761*x^12 +7360*x^13 +5351*x^14 +3496*x^15 +1715*x^16) / (1 -6*x^9 +x^18).

a(9*k-4) = 2401*A001653(k) for k >= 1.

Extensions

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