OFFSET
1,5
COMMENTS
An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABC are equivalent, as are AAABB and BBBAA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a loop are equivalent, so for loops AAABCB = BAAABC = CBAAAB.
FORMULA
T(n,k) = Sum_{j=0..k} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0 <= n <= 1 & n==k].
T(n,k) = Sum_{j=1..k} A304972(n,j).
EXAMPLE
The array begins at T(1,1):
1 1 1 1 1 1 1 1 1 1 1 1 1 ...
1 2 2 2 2 2 2 2 2 2 2 2 2 ...
1 2 3 3 3 3 3 3 3 3 3 3 3 ...
1 4 6 7 7 7 7 7 7 7 7 7 7 ...
1 4 9 11 12 12 12 12 12 12 12 12 12 ...
1 8 18 27 30 31 31 31 31 31 31 31 31 ...
1 8 27 43 55 58 59 59 59 59 59 59 59 ...
1 16 54 107 141 159 163 164 164 164 164 164 164 ...
1 16 81 171 266 312 334 338 339 339 339 339 339 ...
1 32 162 427 688 883 963 993 998 999 999 999 999 ...
1 32 243 683 1313 1774 2069 2169 2204 2209 2210 2210 2210 ...
1 64 486 1707 3407 5103 6119 6634 6789 6834 6840 6841 6841 ...
1 64 729 2731 6532 10368 13524 15080 15790 15975 16026 16032 16033 ...
a(n) are the terms of this array read by antidiagonals.
For T(4,3)=6, the achiral pattern rows are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA. The achiral pattern loops are AAAA, AAAB, AABB, ABAB, AABC, and ABAC.
MATHEMATICA
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] +
Ach[n-2, k-1] + Ach[n-2, k-2]]; (* A304972 *)
Table[Sum[Ach[n, j], {j, 1, k - n + 1}], {k, 1, 15}, {n, 1, k}] // Flatten
CROSSREFS
KEYWORD
AUTHOR
Robert A. Russell, Jun 09 2018
STATUS
approved