A334616 Number of 2-colorings of an n X n X n grid, up to rotational symmetry.

2, 23, 5605504, 768614338020786176, 1772303994379887844373479205703254016, 4388012152856549445746584486819723041078276071004502223505850368, 746581580725934736852480760189481426040548499078234470565449222456544381939194522144498021170453413888

Offset: 1

Author

Paul Oelkers, Sep 08 2020

Keywords

• nonn

Comments

The cycle index of the permutation group is given by:

Even n: (1/24)*(s_1^n^3 + 8*s_1^n*s_3^((n^3-n)/3) + 6*s_2^(n^3/2) + 6*s_4^(n^3/4) + 3*s_2^(n^3/2));

Odd n: (1/24)*(s_1^n^3 + 8*s_1^n*s_3^((n^3-n)/3) + 6*s_1^n*s_2^((n^3-n)/2) + 6*s_1^n*s_4^((n^3-n)/4) + 3*s_1^n*s_2^((n^3-n)/2)).

Examples

			a(2)=23 from:
  00 00
  00 00
------------------------------------------
  10 00
  00 00
------------------------------------------
  11 00   10 00   10 01   10 00
  00 00   01 00   00 00   00 01
------------------------------------------
  11 00   11 00   01 10
  10 00   00 10   10 00
------------------------------------------
  11 00   11 00   01 10   11 00   11 10
  11 00   10 01   10 01   00 11   10 00
------------------------------------------
  00 11   00 11   10 01
  01 11   11 01   01 11
------------------------------------------
  00 11   01 11   01 10   01 11
  11 11   10 11   11 11   11 10
------------------------------------------
  01 11
  11 11
------------------------------------------
  11 11
  11 11
------------------------------------------
An example for the 2-coloring of the 3 X 3 X 3 grid can be written as:
  110 000 111
  100 000 111
  100 000 111
This coloring is equivalent to:
  111 000 111
  001 000 111
  000 000 111
  because you can get this configuration by rotating the first coloring by 90 degrees.
But it is different from:
  011 000 111
  001 000 111
  001 000 111
  because reflections are not considered.
		

Links

• Wikipedia, Cycle index
• Paul Oelkers, Hand written notes and sketches

Crossrefs

This is the three-dimensional version of A047937.

Cf. A000543.

Formula

a(n) = (1/24)*(2^n^3 + 6*2^((n^3)/4) + 9*2^((n^3)/2) + 8*2^((n^3-n)/3+n)) for n even;

a(n) = (1/24)*(2^n^3 + 6*2^(((n^3)-n)/4+n) + 9*2^(((n^3)-n)/2+n) + 8*2^(((n^3-n)/3)+n)) for n odd.

Extensions

More terms from Stefano Spezia, Sep 09 2020