A155923 - OEIS

%I #11 Feb 15 2020 10:52:27

%S 13,17,25,53,85,137,305,493,797,1777,2873,4645,10357,16745,27073,

%T 60365,97597,157793,351833,568837,919685,2050633,3315425,5360317,

%U 11951965,19323713,31242217,69661157,112626853,182092985,406014977,656437405

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

%C (-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.)

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

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

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

%C 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]

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).

%F 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.

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

%e (-5,a(1)) = (-5,13) is a solution: (-5)^2+(-5+17)^2 = 25+144 = 169 = 13^2;

%e (A118120(1), a(2)) = (0, 17) is a solution: 0^2+(0+17)^2 = 289 = 17^2;

%e (A118120(2), a(3)) = (7, 25) is a solution: 7^2+(7+17)^2 = 49+576 = 625 = 25^2.

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

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

%Y 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).

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

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Feb 09 2009

%E G.f. corrected, first and fourth comment and examples edited, cross-reference added by _Klaus Brockhaus_, Sep 22 2009