1,1

(-24, a(1)) and (A130608(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (171+26*sqrt(2))/167 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 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)=145, a(2)=167, a(3)=197, a(4)=673, a(5)=835, a(6)=1037.

G.f.: (1-x)*(145+312*x+509*x^2+312*x^3+145*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 167*A001653(k) for k >= 1.

(-24, a(1)) = (-24, 145) is a solution: (-24)^2+(-24+167)^2 = 576+20449 = 21025 = 145^2.

(A130608(1), a(2)) = (0, 167) is a solution: 0^2+(0+167)^2 = 27889 = 167^2.

(A130608(3), a(4)) = (385, 673) is a solution: 385^2+(385+167)^2 = 148225+304704 = 452929 = 673^2.

(PARI) {forstep(n=-24, 10000000, [1, 3], if(issquare(2*n^2+334*n+27889, &k), print1(k, ", ")))}

Cf. A130608, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).

Sequence in context: A099648 A043652 A164770 this_sequence A051414 A158133 A094613

Adjacent sequences: A159774 A159775 A159776 this_sequence A159778 A159779 A159780

nonn

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