|
%I #4 Dec 25 2017 14:19:58
%S 673,937,1685,2353,4685,9437,13445,27173,54937,78317,158353,320185,
%T 456457,922945,1866173,2660425,5379317,10876853,15506093,31352957,
%U 63394945,90376133,182738425,369492817,526750705,1065077593,2153561957
%N Positive numbers y such that y^2 is of the form x^2+(x+937)^2 with integer x.
%C (-385, a(1)) and (A129974(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+937)^2 = y^2.
%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (1179+506*sqrt(2))/937 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (933747+224782*sqrt(2))/937^2 for n mod 3 = 1.
%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)=673, a(2)=937, a(3)=1685, a(4)=2353, a(5)=4685, a(6)=9437.
%F G.f.: (1-x)*(673+1610*x+3295*x^2+1610*x^3+673*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 937*A001653(k) for k >= 1.
%e (-385, a(1)) = (-385, 673) is a solution: (-385)^2+(-385+937)^2 = 148225+304704 = 452929 = 673^2.
%e (A129974(1), a(2)) = (0, 937) is a solution: 0^2+(0+937)^2 = 877969 = 937^2.
%e (A129974(3), a(4)) = (1128, 2353) is a solution: 1128^2+(1128+937)^2 = 1272384+4264225 = 5536609 = 2353^2.
%t LinearRecurrence[{0,0,6,0,0,-1},{673,937,1685,2353,4685,9437},30] (* _Harvey P. Dale_, Dec 25 2017 *)
%o (PARI) {forstep(n=-388, 10000000, [3, 1], if(issquare(2*n^2+1874*n+877969, &k), print1(k, ",")))}
%Y Cf. A129974, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160210 (decimal expansion of (1179+506*sqrt(2))/937), A160211 (decimal expansion of (933747+224782*sqrt(2))/937^2).
%K nonn
%O 1,1
%A _Klaus Brockhaus_, May 18 2009
|
