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

0,8

Links

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.

Formula

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

Example

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 ...
  ...

Mathematica

{{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 *)

Prog

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

Crossrefs

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.

Sequence in context: A322280 A331436 A343097 * A210472 A320080 A246106

Adjacent sequences: A343092 A343093 A343094 * A343096 A343097 A343098

Keyword

nonn,tabl

Author

Andrew Howroyd, Apr 14 2021

Status

approved