1,1

(-12, a(1)) and (A118675(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (51+14*sqrt(2))/47 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 1.

For the generic case x^2+(x+p)^2=y^2 with p=m^2-2 a prime number in A028871, m>=2, the x values are given by the sequence defined by: a(n)=6*a(n-3)-a(n-6)+2p with a(1)=0, a(2)=2m+2, a(3)=3m^2-10m+8, a(4)=3p, a(5)=3m^2+10m+8, a(6)=20m^2-58m+42.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=m^2+2m+2, b(3)=5m^2-14m+10, b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 09 2009]

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=37, a(2)=47, a(3)=65, a(4)=157, a(5)=235, a(6)=353.

G.f.: (1-x)*(37+84*x+149*x^2+84*x^3+37*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 47*A001653(k) for k >= 1.

(-12, a(1)) = (-12, 37) is a solution: (-12)^2+(-12+47)^2 = 144+1225 = 1369 = 37^2.

(A118675(1), a(2)) = (0, 47) is a solution: 0^2+(0+47)^2 = 2209 = 47^2.

(A118675(3), a(4)) = (85, 157) is a solution: 85^2+(85+47)^2 = 7225+17424 = 24649 = 157^2.

(PARI) {forstep(n=-12, 100000000, [1, 3], if(issquare(2*n^2+94*n+2209, &k), print1(k, ", ")))}

Cf. A118675, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).

Sequence in context: A039351 A043174 A043954 this_sequence A108333 A039465 A083240

Adjacent sequences: A159747 A159748 A159749 this_sequence A159751 A159752 A159753

nonn

Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 30 2009