A244094 -id:A244094 - cp's OEIS frontend

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

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}]

A244093 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.

Original entry on oeis.org

18, 11, 11, 12, 13, 15, 17, 19, 22, 25, 28, 31, 35, 39, 42, 47, 51, 56, 60, 65, 70, 76, 81, 87, 93, 99, 106, 112, 119, 126, 133, 141, 148, 156, 164, 173, 181, 190, 198, 207, 217, 226, 236, 246, 256, 266, 276, 287, 298, 309, 320, 332, 343, 355, 367, 380, 392, 405, 418, 431, 444

Offset: 3

Kival Ngaokrajang, Jun 20 2014

nonn

The minimum ratio occurs at n = 5.

Kival Ngaokrajang, Illustration of initial terms

Cf. A244094, A244096.

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

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]

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

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

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

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A244093 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.

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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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Author

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Crossrefs

Programs

PARI

Formula