A157247 - OEIS

%I #15 Mar 17 2023 07:14:09

%S 1715,1781,1855,2009,2401,2989,3451,3821,4459,5831,6865,7679,9065,

%T 12005,15925,18851,21145,25039,33271,39409,44219,52381,69629,92561,

%U 109655,123049,145775,193795,229589,257635,305221,405769,539441,639079,717149

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

%C (-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.

%C lim_{n -> infinity} a(n)/a(n-9) = 3+2*sqrt(2).

%C 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}.

%C 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}.

%C 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}.

%F 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.

%F 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).

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

%e (-1029, a(1)) = (-1029, 1715) is a solution: (-1029)^2+(-1029+2401)^2 = 1058841+1882384 = 2941225 = 1715^2.

%e (A118630(1), a(5)) = (0, 2401) is a solution: 0^2+(0+2401)^2 = 5764801 = 2401^2.

%e (A118630(3), a(7)) = (924, 3451) is a solution: 924^2+(924+2401)^2 = 853776+11055625 = 11909401 = 3451^2.

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

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

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

%K nonn

%O 1,1

%A _Klaus Brockhaus_, Feb 25 2009

%E G.f. adapted to the offset by _Bruno Berselli_, Apr 01 2011