1,1

(-56, a(1)) and (A130647(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (843+58*sqrt(2))/839 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 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)=785, a(2)=839, a(3)=901, a(4)=3809, a(5)=4195, a(6)=4621.

G.f.: (1-x)*(785+1624*x+2525*x^2+1624*x^3+785*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 839*A001653(k) for k >= 1.

(-56, a(1)) = (-56, 785) is a solution: (-56)^2+(-56+839)^2 = 3136+613089 = 616225 = 785^2.

(A130647(1), a(2)) = (0, 839) is a solution: 0^2+(0+839)^2 = 703921 = 839^2.

(A130647(3), a(4)) = (2241, 3809) is a solution: 2241^2+(2241+839)^2 = 5022081+9486400 = 14508481 = 3809^2.

(PARI) {forstep(n=-56, 10000000, [1, 3], if(issquare(2*n^2+1678*n+703921, &k), print1(k, ", ")))}

Cf. A130647, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159897 (decimal expansion of (843+58*sqrt(2))/839), A159898 (decimal expansion of (1760979+1141390*sqrt(2))/839^2).

Sequence in context: A146978 A095954 A151658 this_sequence A031734 A031616 A097776

Adjacent sequences: A159893 A159894 A159895 this_sequence A159897 A159898 A159899

nonn

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