A244093 -id:A244093 - cp's OEIS frontend

A244094 Rounded down ratio of area of a unit circle and one of the circles inscribed between a regular n-gon and a circumscribed unit circle.

16, 46, 109, 222, 407, 690, 1099, 1669, 2437, 3445, 4737, 6363, 8376, 10834, 13797, 17330, 21503, 26389, 32064, 38608, 46108, 54651, 64331, 75243, 87489, 101172, 116402, 133290, 151953, 172511, 195088, 219813, 246817, 276236, 308211, 342885, 380407, 420927, 464601, 511590

Offset: 3

Kival Ngaokrajang, Jun 20 2014

Kival Ngaokrajang, Illustration of initial terms

Cf. A244093, A244096.

{
  for (n=3, 100,
     b=Pi*(n-2)/(2*n);
     r=(1-sin(b))/2;
     area=Pi*r^2;
     a=floor(Pi/area);
     print1(a,", ")
  )
}

a(n) = floor(Pi/area(n)) where area(n) = Pi*r(n)^2, r(n) = (1 - cos(Pi/n))/2 = sin(Pi/(2*n))^2, n >= 3.

Edited. Formula simplified - Wolfdieter Lang, Jun 27 2014

A234924 Combined weight, as defined at A244094, of the distinct-parts partitions of n.

Original entry on oeis.org

1, 2, 8, 11, 22, 41, 60, 89, 136, 208, 275, 397, 526, 724, 978, 1279, 1646, 2172, 2752, 3518, 4492, 5620, 7010, 8742, 10809, 13280, 16346, 19937, 24200, 29373, 35436, 42548, 51153, 61039, 72794, 86632, 102615, 121268, 143209, 168458, 197753, 231833, 270983

Offset: 1

Clark Kimberling, Jan 01 2014

These are the row sums of the array at A234923.

Cf. A244093, A234097.

z = 45; p[n_] := p[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; q[n_] := q[n] = Length[p[n]]; v[n_] := v[n] = Table[n + 1 - i, {i, 1, n}]; w[n_, h_] := w[n, h] = Dot[p[n][[h]], v[Length[p[n][[h]]]]]; a[n_] := Sum[w[n, h], {h, 1, q[n]}]; Table[a[n], {n, 1, z}]

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.

Original entry on oeis.org

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

Kival Ngaokrajang, Jun 20 2014

nonn

Kival Ngaokrajang, Illustration of initial terms

Cf. A244093, A244094.

{
  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,", ")
  )
}

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]

cp's OEIS Frontend

A244094 Rounded down ratio of area of a unit circle and one of the circles inscribed between a regular n-gon and a circumscribed unit circle.

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A234924 Combined weight, as defined at A244094, of the distinct-parts partitions of n.

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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.

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