A108914 Number of regions formed inside square by diagonals and the segments joining the vertices to the points dividing the sides into n equal length segments.

4, 32, 96, 188, 332, 460, 712, 916, 1204, 1488, 1904, 2108, 2716, 3080, 3532, 4068, 4772, 5140, 6016, 6392, 7188, 7992, 8936, 9260, 10484, 11312, 12208, 12968, 14396, 14660, 16504, 17220, 18436, 19680, 20756, 21548, 23692, 24728, 25992, 26868, 29204, 29704, 32176, 33068, 34444, 36552, 38552

Offset: 1

Author

Len Smiley and Brian Wick ( mathclub(AT)math.uaa.alaska.edu ), Jul 19 2005

Keywords

• nonn

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..100
• Scott R. Shannon, Image for n = 2.
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 6.
• Scott R. Shannon, Image for n = 11.
• Scott R. Shannon, Image for n = 30.
• L. Smiley, The case n=6. Note 3- and 4-fold off-diagonal concurrencies.
• L. Smiley, The case n=7. Note there are no off-diagonal concurrencies.

Crossrefs

A092098 is the corresponding count for triangles.

A355949 (vertices), A355948 (edges), A355992 (k-gons), A355838, A355798.

Formula

If n=1 or n is prime, a(n)=18*n^2-26*n+12.

If n is composite, vanishing regions from 3- and 4-fold concurrency must be subtracted.

a(n) = A355948(n) - A355949(n) + 1 by Euler's formula.

Extensions

a(23), a(33) corrected, a(41) and above by Scott R. Shannon, Jul 22 2022