1,1

(-52, a(1)) and (A130646(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (731+54*sqrt(2))/727 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 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>=5, 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)=677, a(2)=727, a(3)=785, a(4)=3277, a(5)=3635, a(6)=4033.

G.f.: (1-x)*(677+1404*x+2189*x^2+1404*x^3+677*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 727*A001653(k) for k >= 1.

(-52, a(1)) = (-52, 677) is a solution: (-52)^2+(-52+727)^2 = 2704+455625 = 458329 = 677^2.

(A130646(1), a(2)) = (0, 727) is a solution: 0^2+(0+727)^2 = 528529 = 727^2.

(A130646(3), a(4)) = (1925, 3277) is a solution: 1925^2+(1925+727)^2 = 3705625+7033104 = 10738729 = 3277^2.

(PARI) {forstep(n=-52, 10000000, [1, 3], if(issquare(2*n^2+1454*n+528529, &k), print1(k, ", ")))}

Cf. A130646, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).

Sequence in context: A014759 A058462 A058450 this_sequence A142755 A158386 A031614

Adjacent sequences: A159890 A159891 A159892 this_sequence A159894 A159895 A159896

nonn

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