A244096 Rounded down area ratio of a circle inscribed in a congruent triangle of a regular n-gon and a circle inscribed between a side of such an n-gon and a circumscribed unit circle.

0, 4, 9, 18, 30, 45, 63, 84, 108, 135, 166, 200, 237, 277, 321, 367, 417, 471, 527, 587, 649, 716, 785, 858, 933, 1012, 1095, 1180, 1269, 1361, 1456, 1555, 1656, 1761, 1870, 1981, 2096, 2214, 2335, 2459, 2587, 2718, 2852, 2989, 3130, 3274, 3421, 3571, 3725, 3881, 4042

Offset: 3

Author

Kival Ngaokrajang, Jun 20 2014

Keywords

• nonn

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A244093, A244094.

Programs

PARI

{
  for (n=3, 100,
     c=2*sin(Pi/n);
     s=(2+c)/2;
     r1=(((s-1)^2*(s-c))/s)^(1/2);
     b=Pi*(n-2)/(2*n);
     r2=(1-sin(b))/2;
     a=floor(r1^2/r2^2);
     print1(a,", ")
  )
}

Formula

a(n) = floor((r1(n)/r2(n))^2) where r1(n) = (s(n)/2)*sqrt((2 - s(n))/(2 + s(n))) and r2(n) = (2 - c(n))/4 with s(n) = 2*sin(Pi/n), the side length (length unit 1), and c(n) = 2*cos(Pi/n), the length ratio of the smallest diagonal and the side of a regular n-gon. [Rewritten by Wolfdieter Lang, Jul 02 2014]