1,1

(-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}.

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.: (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.

(-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.

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

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

Sequence in context: A062916 A129540 A008744 this_sequence A024408 A140910 A025039

Adjacent sequences: A157244 A157245 A157246 this_sequence A157248 A157249 A157250

nonn

Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Feb 25 2009