%I #11 Apr 18 2024 12:53:46

%S 313,337,365,1513,1685,1877,8765,9773,10897,51077,56953,63505,297697,

%T 331945,370133,1735105,1934717,2157293,10112933,11276357,12573625,

%U 58942493,65723425,73284457,343542025,383064193,427133117,2002309657

%N Positive numbers y such that y^2 is of the form x^2+(x+337)^2 with integer x.

%C (-25,a(1)) and (A129999(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+337)^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)=313, a(2)=337, a(3)=365, a(4)=1513, a(5)=1685, a(6)=1877.

%F G.f.: x*(1-x)*(313+650*x+1015*x^2+650*x^3+313*x^4) / (1-6*x^3+x^6).

%F a(3*k-1) = 337*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) = (339+26*sqrt(2))/337 for n mod 3 = {0, 2}.

%F Limit_{n -> oo} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 1.

%e (-25, a(1)) = (-25, 313) is a solution: (-25)^2+(-25+337)^2 = 625+97344 = 97969 = 313^2.

%e (A129993(1), a(2)) = (0, 337) is a solution: 0^2+(0+337)^2 = 113569 = 337^2.

%e (A129993(3), a(4)) = (888, 1513) is a solution: 888^2+(888+337)^2 = 788544+1500625 = 2289169 = 1513^2.

%o (PARI) {forstep(n=-28, 50000000, [3, 1], if(issquare(2*n^2+674*n+113569, &k), print1(k, ",")))}

%Y Cf. A129999, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576(decimal expansion of (278307+179662*sqrt(2))/337^2).

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Apr 16 2009