cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A217746 Number of polygonal regions with finite area in the exterior of a regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 0, 7, 24, 63, 120, 242, 384, 650, 896, 1425, 1872, 2703
Offset: 3

Views

Author

Martin Renner, Mar 23 2013

Keywords

Examples

			a(7) = 7 since the 28 diagonals of the regular heptagon divide the exterior in 35 regions consisting of seven triangles (with finite area), i.e., 1 triangle (7 times), and 28 regions with infinite area of three different shapes (two 7 times, one 14 times).
a(8) = 24 since the 40 diagonals of the regular octagon divide the exterior in 64 regions consisting of 24 polygons (with finite area), i.e., 2 triangles (one 8 times, one 16 times), and 40 regions with infinite area of three different shapes (one 8 times, two 16 times).
a(9) = 63 since the 54 diagonals of the regular 9-gon (nonagon) divide the exterior in 117 regions consisting of 63 polygons (with finite area), i.e., 3 triangles (one 9 times, two 18 times) and 2 quadrilaterals (each 9 times), and 54 regions with infinite area of four different shapes (two 9 times, two 18 times).

Crossrefs

Cf. A007678, A217745, A217748.

Formula

a(n) = A217745(n) - A217748(n)

A217748 Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068, 2160, 2254, 2350, 2448, 2548
Offset: 3

Views

Author

Martin Renner, Mar 23 2013

Keywords

Comments

For n > 3 same as A028552(n-3).

Examples

			a(3) = 1 since the equilateral triangle has no diagonals and therefore one exterior region with infinite area.
a(4) = 4 since the two diagonals of the square divide the exterior in four regions with infinite area.
a(5) = 10 since the ten diagonals of the regular pentagon divide the exterior in ten regions with infinite area of two different shapes.

Crossrefs

Cf. A004526, A007678, A028552, A164004, A217745, A217746.

Programs

  • Mathematica
    a[n_] := n*(n - 3); a[3] = 1; Array[a, 50, 3] (* Amiram Eldar, Dec 10 2022 *)
  • PARI
    a(n) = if(n == 3, 1, n*(n-3)); \\ Amiram Eldar, Dec 10 2022

Formula

a(n) = n*(n-3) for n > 3.
a(n) = A217745(n) - A217746(n).
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=3} 1/a(n) = 29/18.
Sum_{n>=3} (-1)^(n+1)/a(n) = 23/18 - 2*log(2)/3. (End)

A217753 Number of noncongruent polygonal regions with finite area in the exterior of a regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 7, 14, 18, 30, 35, 55, 62, 90
Offset: 3

Views

Author

Martin Renner, Mar 23 2013

Keywords

Examples

			a(7) = 1 since the 35 exterior regions of the regular heptagon built by all diagonals consist of one noncongruent polygon, i.e., 1 triangle (7 times), and three different noncongruent regions with infinite area (two 7 times, one 14 times).
a(8) = 2 since the 64 exterior regions of the regular octagon built by all diagonals consist of two different noncongruent polygons, i.e., 2 triangles (one 8 times, one 16 times), and three different noncongruent regions with infinite area (one 8 times, two 16 times).
a(9) = 5 since the 117 exterior regions of the regular 9-gon (nonagon) built by all diagonals consist of five different noncongruent polygons, i.e., 3 triangles (one 9 times, two 18 times) and 2 quadrilaterals (each 9 times), and four different noncongruent regions with infinite area (two 9 times, two 18 times).

Crossrefs

Cf. A004526, A187781, A217745, A217746, A217748, A217754.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.