A357216 - OEIS

A357216

Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

1, 4, 1, 13, 5, 1, 28, 17, 6, 1, 49, 37, 21, 7, 1, 70, 65, 46, 25, 8, 1, 109, 93, 81, 55, 29, 9, 1, 148, 145, 126, 97, 64, 33, 10, 1, 181, 181, 181, 151, 113, 73, 37, 11, 1, 244, 257, 246, 217, 176, 129, 82, 41, 12, 1, 301, 309, 321, 295, 253, 201, 145, 91, 45, 13, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

3,2

COMMENTS

Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon).

LINKS

Table of n, a(n) for n=3..68.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.

Scott R. Shannon, Image of T(5,20) = 2001.

Scott R. Shannon, Image of T(7,10) = 701.

FORMULA

T(n,k) = A357254(n,k) - A357235(n,k) + 1 by Euler's formula.

T(n,0) = 1.

T(n,1) = n + 1.

Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + 1.

T(3,k) = A356984(k).

T(4,k) = A357058(k).

T(6,k) = A357196(k).

Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + 1.

EXAMPLE

The table begins:

1, 4, 13, 28, 49, 70, 109, 148, 181, 244, 301, 334, 433, 508, 565, ...

1, 5, 17, 37, 65, 93, 145, 181, 257, 309, 401, 457, 577, 653, 785, ...

1, 6, 21, 46, 81, 126, 181, 246, 321, 406, 501, 606, 721, 846, 981, ...

1, 7, 25, 55, 97, 151, 217, 295, 385, 475, 601, 715, 865, 1015, 1159, ...

1, 8, 29, 64, 113, 176, 253, 344, 449, 568, 701, 848, 1009, 1184, 1373, ...

1, 9, 33, 73, 129, 201, 289, 393, 513, 649, 801, 969, 1153, 1353, 1569, ...

1, 10, 37, 82, 145, 226, 325, 442, 577, 730, 901, 1090, 1297, 1522, 1765, ...

1, 11, 41, 91, 161, 251, 361, 491, 641, 811, 1001, 1211, 1441, 1691, 1961, ...

1, 12, 45, 100, 177, 276, 397, 540, 705, 892, 1101, 1332, 1585, 1860, 2157, ...

1, 13, 49, 109, 193, 301, 433, 589, 769, 973, 1201, 1453, 1729, 2029, 2353, ...

1, 14, 53, 118, 209, 326, 469, 638, 833, 1054, 1301, 1574, 1873, 2198, 2549, ...

1, 15, 57, 127, 225, 351, 505, 687, 897, 1135, 1401, 1695, 2017, 2367, 2745, ...

1, 16, 61, 136, 241, 376, 541, 736, 961, 1216, 1501, 1816, 2161, 2536, 2941, ...

...

See the attached text file for further examples.

See A356984, A357058, A357196 for more images of the n-gons.

CROSSREFS

Cf. A357235 (vertices), A357254 (edges), A356984 (triangle), A357058 (square), A357196 (hexagon), A007678, A344857.

Sequence in context: A303547 A184753 A324186 * A055252 A318945 A193956

Adjacent sequences: A357213 A357214 A357215 * A357217 A357218 A357219

KEYWORD

nonn,tabl

AUTHOR

Scott R. Shannon, Sep 18 2022

STATUS

approved