A155923 Positive numbers y such that y^2 is of the form x^2+(x+17)^2 with integer x.

13, 17, 25, 53, 85, 137, 305, 493, 797, 1777, 2873, 4645, 10357, 16745, 27073, 60365, 97597, 157793, 351833, 568837, 919685, 2050633, 3315425, 5360317, 11951965, 19323713, 31242217, 69661157, 112626853, 182092985, 406014977, 656437405

Offset: 1

Author

Klaus Brockhaus, Feb 09 2009

Keywords

• nonn
• easy

Comments

(-5,a(1)) and (A118120(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2. (Offset 1 is assumed for A118120.)

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

lim_{n -> infinity} a(n)/a(n-1) = (19+6*sqrt(2))/17 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (387+182*sqrt(2))/17^2 for n mod 3 = 1.

For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, 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+1, a(3)=6m^2-10m+4, a(4)=3p, a(5)=6m^2+10m+4, a(6)=40m^2-58m+21.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=2*m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. [From Mohamed Bouhamida, Sep 09 2009]

Examples

			(-5,a(1)) = (-5,13) is a solution: (-5)^2+(-5+17)^2 = 25+144 = 169 = 13^2;
(A118120(1), a(2)) = (0, 17) is a solution: 0^2+(0+17)^2 = 289 = 17^2;
(A118120(2), a(3)) = (7, 25) is a solution: 7^2+(7+17)^2 = 49+576 = 625 = 25^2.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A118120, A156035 (decimal expansion of 3+2*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17), A157649 (decimal expansion of (387+182*sqrt(2))/17^2).

Cf. A156156 (first trisection), A156157 (second trisection), A156158 (third trisection).

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{13,17,25,53,85,137},50] (* Harvey P. Dale, Feb 11 2015 *)

PARI

{forstep(n=-5, 660000000, [1,3], if(issquare(2*n*(n+17)+289, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1) = 13, a(2) = 17, a(3) = 25, a(4) = 53, a(5) = 85, a(6) = 137.

G.f.: x*(1-x)*(13+30*x+55*x^2+30*x^3+13*x^4)/(1-6*x^3+x^6).

Extensions

G.f. corrected, first and fourth comment and examples edited, cross-reference added by Klaus Brockhaus, Sep 22 2009