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A305749
T(n,k) is the number of achiral color patterns (set partitions) in a row or loop of length n with k or fewer colors (sets).
10
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 18, 8, 1, 1, 2, 3, 7, 12, 27, 27, 16, 1, 1, 2, 3, 7, 12, 30, 43, 54, 16, 1, 1, 2, 3, 7, 12, 31, 55, 107, 81, 32, 1, 1, 2, 3, 7, 12, 31, 58, 141, 171, 162, 32, 1, 1, 2, 3, 7, 12, 31, 59, 159, 266, 427, 243, 64, 1, 1, 2, 3, 7, 12, 31, 59, 163, 312, 688, 683, 486, 64, 1
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
Columns 1-6 are A057427, A016116, A182522, A305750, A305751, and A305752.
Columns converge to the right to A080107.
Sequence in context: A323767 A357824 A159936 * A320748 A320747 A348690
KEYWORD
nonn,tabl,easy
AUTHOR
Robert A. Russell, Jun 09 2018
STATUS
approved