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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.
3
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
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).
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
KEYWORD
nonn,tabl
AUTHOR
Scott R. Shannon, Sep 18 2022
STATUS
approved