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Rounded down ratio of area of a unit circle and a circle inscribed in any of the n triangles composing a regular n-gon which is circumscribed by a unit circle.
4

%I #22 Sep 16 2019 12:40:28

%S 18,11,11,12,13,15,17,19,22,25,28,31,35,39,42,47,51,56,60,65,70,76,81,

%T 87,93,99,106,112,119,126,133,141,148,156,164,173,181,190,198,207,217,

%U 226,236,246,256,266,276,287,298,309,320,332,343,355,367,380,392,405,418,431,444

%N Rounded down ratio of area of a unit circle and a circle inscribed in any of the n triangles composing a regular n-gon which is circumscribed by a unit circle.

%C The minimum ratio occurs at n = 5.

%H Kival Ngaokrajang, <a href="/A244093/a244093_2.pdf">Illustration of initial terms</a>

%F a(n) = floor(Pi/area(n)) where area = Pi*r(n)^2, r(n) = (s(n)/2)*sqrt((2 - s(n))/(2 + s(n))), with s(n) = 2*sin(Pi/n) which is the side length (length unit 1) of the regular n gon. [rewritten by _Wolfdieter Lang_, Jun 30 2014 and Jul 02 2014]

%F a(n) = floor(1/r(n)^2) with r(n) = S(n)*(1 + C(n) - S(n))/(1 + C(n) + S(n)) with S(n) = s(n)/2 and C(n) = cos(Pi/n). 2*C(n) is the ratio of the length of the smallest diagonal and the side length s(n) in the regular n-gon. - _Wolfdieter Lang_, Jun 30 2014

%o (PARI)

%o {

%o for (n=3, 100,

%o c=2*sin(Pi/n);

%o s=(2+c)/2;

%o r=sqrt(((s-1)^2*(s-c))/s);

%o area=Pi*r^2;

%o a=floor(Pi/area);

%o print1(a,", ")

%o )

%o }

%Y Cf. A244094, A244096.

%K nonn

%O 3,1

%A _Kival Ngaokrajang_, Jun 20 2014