%I #12 Apr 18 2024 11:00:21

%S 97,137,277,305,685,1565,1733,3973,9113,10093,23153,53113,58825,

%T 134945,309565,342857,786517,1804277,1998317,4584157,10516097,

%U 11647045,26718425,61292305,67883953,155726393,357237733,395656673,907639933

%N Positive numbers y such that y^2 is of the form x^2+(x+137)^2 with integer x.

%C (-65, a(1)) and (A129544(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

%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)=97, a(2)=137, a(3)=277, a(4)=305, a(5)=685, a(6)=1565.

%F G.f.: x*(1-x)*(97+234*x+511*x^2+234*x^3+97*x^4)/(1-6*x^3+x^6).

%F a(3*k-1) = 137*A001653(k) for k >= 1.

%F Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).

%F Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 1.

%F Limit_{n -> oo} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}.

%e (-65, a(1)) = (-65, 97) is a solution: (-65)^2+(-65+137)^2 = 4225+5184 = 9409 = 97^2.

%e (A129544(1), a(2)) = (0, 137) is a solution: 0^2+(0+137)^2 = 18769 = 137^2.

%e (A129544(3), a(4)) = (136, 305) is a solution: 136^2+(136+137)^2 = 18496+74529 = 93025 = 305^2.

%o (PARI) {forstep(n=-68, 1000000000, [3, 1], if(issquare(n^2+(n+137)^2,&k), print1(k, ",")))}

%Y Cf. A129544, A001653, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))).

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Feb 25 2009