A047939 -id:A047939 - cp's OEIS frontend

A343095 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotational symmetry.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 24, 140, 1, 0, 1, 5, 70, 4995, 16456, 1, 0, 1, 6, 165, 65824, 10763361, 8390720, 1, 0, 1, 7, 336, 489125, 1073758336, 211822552035, 17179934976, 1, 0, 1, 8, 616, 2521476, 38147070625, 281474993496064, 37523658921114744, 140737496748032, 1, 0

Offset: 0

Andrew Howroyd, Apr 14 2021

			Array begins:
====================================================================
n\k | 0 1       2            3               4                 5
----+---------------------------------------------------------------
  0 | 1 1       1            1               1                 1 ...
  1 | 0 1       2            3               4                 5 ...
  2 | 0 1       6           24              70               165 ...
  3 | 0 1     140         4995           65824            489125 ...
  4 | 0 1   16456     10763361      1073758336       38147070625 ...
  5 | 0 1 8390720 211822552035 281474993496064 74505806274453125 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..860
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.

Rows 0..5 are A000012, A001477, A006528, A282613, A283027, A283031.
Columns 0..10 are A000007, A000012, A047937, A047938, A047939, A047940, A047941, A047942, A047943, A047944, A047945.
Main diagonal is A343096.
Cf. A182406, A246106, A343097, A343874.

{{1}}~Join~Table[Function[n, (k^(n^2) + 2*k^((n^2 + 3 #)/4) + k^((n^2 + #)/2))/4 &[Mod[n, 2] ] ][m - k + 1], {m, 0, 8}, {k, m + 1, 0, -1}] // Flatten (* Michael De Vlieger, Nov 30 2023 *)

T(n,k) = (k^(n^2) + 2*k^((n^2 + 3*(n%2))/4) + k^((n^2 + (n%2))/2))/4

T(n,k) = (k^(n^2) + 2*k^((n^2 + 3*(n mod 2))/4) + k^((n^2 + (n mod 2))/2))/4.

A047937 Number of 2-colorings of an n X n grid, up to rotational symmetry.

Original entry on oeis.org

1, 2, 6, 140, 16456, 8390720, 17179934976, 140737496748032, 4611686019501162496, 604462909807864344215552, 316912650057057631849169289216, 664613997892457937028364283517337600, 5575186299632655785385110159782842147536896, 187072209578355573530071668259090783437390809661440

Offset: 0

Rob Pratt

Cycle index = 1/4(s_1^(n^2)+ 2 s_4^floor(n^2/4)s_1^(n mod 2)+s_2^floor(n^2/2)s_1^(n mod 2)). - Geoffrey Critzer, Oct 28 2011

			a(2)=6 from
00 10 11 10 11 11
00 00 00 01 10 11

Andrew Howroyd, Table of n, a(n) for n = 0..50
Peter Kagey, Illustration of a(3)=140
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Column k=2 of A343095.
Other columns are A047938, A047939, A047940, A047941, A047942, A047943, A047944, A047945.
Cf. A054247.

Table[(2^(n^2)+2*2^Floor[n^2/4]*2^Mod[n,2]+2^Floor[n^2/2]*2^Mod[n,2])/4,{n,0,10}]  (* Geoffrey Critzer, Oct 28 2011 *)

a(n) = (m^(n^2) + 2*m^((n^2 + 3*(n mod 2))/4) + m^((n^2 + (n mod 2))/2))/4, with m = 2.

Terms a(12) and beyond from Andrew Howroyd, Apr 14 2021

A047938 Number of ways to label points of an n X n grid with 3 colors, up to rotational symmetry.

Original entry on oeis.org

1, 3, 24, 4995, 10763361, 211822552035, 37523658921114744, 59824832307866205347043, 858420955073128584419531008641, 110856622060759442496180656754310346403, 128844380183002832759115461915902241562318377784, 1347757724935823407809884872163997148505019182125807296675

Offset: 0

Rob Pratt

nonn

Labels of adjacent points need not be distinct.

Andrew Howroyd, Table of n, a(n) for n = 0..25

Column k=3 of A343095.
See A068253 for number of proper colorings.
Cf. A054739, A047937, A047939.

a(n) = (m^(n^2) + 2 m^((n^2 + 3 (n mod 2))/4) + m^((n^2 + (n mod 2))/2))/4, with m = 3.

Terms a(10) and beyond from Andrew Howroyd, Apr 14 2021

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A343095 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotational symmetry.

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A047937 Number of 2-colorings of an n X n grid, up to rotational symmetry.

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A047938 Number of ways to label points of an n X n grid with 3 colors, up to rotational symmetry.

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