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

1,1

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.

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.

Crossrefs

A092098 is the corresponding count for triangles.

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

Sequence in context: A153794 A222326 A370082 * A052469 A211625 A211630

Adjacent sequences: A108911 A108912 A108913 * A108915 A108916 A108917

Keyword

nonn

Author

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

Extensions

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

Status

approved