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

1,1

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}.

Links

Table of n, a(n) for n=1..35.

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.

Example

(-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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A008744 A184090 A297560 * A347969 A267201 A024408

Adjacent sequences: A157244 A157245 A157246 * A157248 A157249 A157250

Keyword

nonn

Author

Klaus Brockhaus, Feb 25 2009

Extensions

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

Status

approved