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

3,2

Links

Table of n, a(n) for n=3..53.

Kival Ngaokrajang, Illustration of initial terms

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]

Prog

(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, ", ")
  )
}

Crossrefs

Cf. A244093, A244094.

Sequence in context: A008025 A301196 A008020 * A008146 A038098 A299274

Adjacent sequences: A244093 A244094 A244095 * A244097 A244098 A244099

Keyword

nonn

Author

Kival Ngaokrajang, Jun 20 2014

Status

approved