BBC Russian

Svoboda | Graniru | BBC Russia | Golosameriki | Facebook

The OEIS is supported by the many generous donors to the OEIS Foundation.

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) = nk^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) = nk^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`

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 29 16:10 EDT 2024. Contains 375517 sequences. (Running on oeis4.)