A356799 - OEIS

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A356799

Table read by antidiagonals: T(n,k) (n >= 2, k >= 1) is the number of regions formed in a regular 2n-gon by straight line segments when connecting the k+1 points that divide each side into k equal parts to the equivalent point on the side diagonally opposite.

1

1, 4, 13, 9, 24, 25, 16, 55, 48, 41, 25, 66, 105, 70, 61, 36, 121, 144, 171, 108, 85, 49, 126, 233, 220, 253, 140, 113, 64, 211, 288, 381, 312, 351, 192, 145, 81, 204, 409, 450, 565, 448, 465, 234, 181, 100, 325, 480, 671, 636, 785, 608, 595, 300, 221, 121, 300, 633, 760, 997, 924, 1041, 738, 741, 352, 265

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OFFSET

2,2

COMMENTS

Many rows and columns in the table appear to be given by a quadratic in even and odd values of k and n; see the Formula section. The exceptions are for rows with n mod 6 = 0 for even k, and for columns with even k, formulas for which are unknown.

LINKS

Table of n, a(n) for n=2..67.

Scott R. Shannon, Table for n=2..35, k=1..50.

Scott R. Shannon, Image for T(2,8) = 64.

Scott R. Shannon, Image for T(3,6) = 126.

Scott R. Shannon, Image for T(3,7) = 211.

Scott R. Shannon, Image for T(4,8) = 480.

Scott R. Shannon, Image for T(4,9) = 633.

Scott R. Shannon, Image for T(6,12) = 2232.

Scott R. Shannon, Image for T(6,13) = 3013.

Scott R. Shannon, Image for T(10,10) = 5100.

Scott R. Shannon, Image for T(10,11) = 6581.

Scott R. Shannon, Image for T(18,1) = 613.

Scott R. Shannon, Image for T(18,2) = 972.

Scott R. Shannon, Image for T(18,3) = 2485.

Scott R. Shannon, Image for T(18,4) = 3096.

Scott R. Shannon, Image for T(18,5) = 5581.

Scott R. Shannon, Image for T(18,6) = 6444.

FORMULA

T(2,k) = k^2.

Conjectured formula for the rows for odd values of k for n>=3:

T(n,k) = A000217(n-1)*k^2 + n^2*k + A000217(n-2) = (n^2 - n)*k^2/2 + n^2*k + (n^2 - 3n + 2)/2.

E.g., T(7,k) = A000217(6)*k^2 + 7^2*k + A000217(5) = 21k^2 + 49k + 15.

Conjectured formula for the rows for even values of k for n>=3:

For n mod 3 = 1 or n mod 3 = 2, T(n,k) = A000217(n-1)*k^2 + A265225(n-1)*k = (n^2 - n)*k^2/2 + (floor(n/2) + 1)*n*k.

E.g., T(10,k) = A000217(9)*k^2 + A265225(9)*k = 45k^2 + 60k.

For n mod 6 = 0, no formula is currently known.

For (n - 3) mod 6 = 0, T(n,k) = A000096(2n-3)*k^2/4 + A005563(n)*k/2 = (2n^2 - 3n)*k^2/4 + (n^2 + 2n)*k/2.

E.g., T(15,k) = 405k^2/4 + 255k/2.

Conjectured formula for the columns for odd values of k for n>=3:

T(n,k) = A001105((k+1)/2)*n^2 - A051890((k+1)/2)*n + 1 = (k^2 + 2k + 1)*n^2/2 - (k^2 + 3)*n/2 + 1.

E.g., T(n,9) = 50n^2 - 42n + 1.

Conjectured formula for T(n,2):

T(n,2) = 2*A249127(n) = 2*floor(3n/2)*n, for n>=3.

No formula is current known for the columns for even values of k for k>=4.

EXAMPLE

The table begins:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...

13, 24, 55, 66, 121, 126, 211, 204, 325, 300, 463, 414, ...

25, 48, 105, 144, 233, 288, 409, 480, 633, 720, 905, 1008, ...

41, 70, 171, 220, 381, 450, 671, 760, 1041, 1150, 1491, 1620, ...

61, 108, 253, 312, 565, 636, 997, 1056, 1549, 1596, 2221, 2232, ...

85, 140, 351, 448, 785, 924, 1387, 1568, 2157, 2380, 3095, 3360, ...

113, 192, 465, 608, 1041, 1248, 1841, 2112, 2865, 3200, 4113, 4512, ...

145, 234, 595, 738, 1333, 1512, 2359, 2556, 3673, 3870, 5275, 5454, ...

181, 300, 741, 960, 1661, 1980, 2941, 3360, 4581, 5100, 6581, 7200, ...

221, 352, 903, 1144, 2025, 2376, 3587, 4048, 5589, 6160, 8031, 8712, ...

265, 432, 1081, 1344, 2425, 2784, 4297, 4704, 6697, 7152, 9625, 10080, ...

313, 494, 1275, 1612, 2861, 3354, 5071, 5720, 7905, 8710, 11363, 12324, ...

365, 588, 1485, 1904, 3333, 3948, 5909, 6720, 9213, 10220, 13245, 14448, ...

421, 660, 1711, 2130, 3841, 4410, 6811, 7500, 10621, 11400, 15271, 16110, ...

481, 768, 1953, 2496, 4385, 5184, 7777, 8832, 12129, 13440, 17441, 19008, ...

545, 850, 2211, 2788, 4965, 5814, 8807, 9928, 13737, 15130, 19755, 21420, ...

613, 972, 2485, 3096, 5581, 6444, 9901, 10944, 15445, 16668, 22213, 23544, ...

.

.

CROSSREFS

Cf. A000217, A265225, A000096, A000290, A005563, A001844, A001105, A051890, A249127, A356044.

Sequence in context: A046738 A095324 A264341 * A144290 A360390 A101181

Adjacent sequences: A356796 A356797 A356798 * A356800 A356801 A356802

KEYWORD

nonn,tabl

AUTHOR

Scott R. Shannon, Aug 28 2022

STATUS

approved