This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A305749 #7 Jul 28 2018 11:41:38
%S A305749 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,
%T A305749 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,
%U A305749 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
%N 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).
%C A305749 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.
%F A305749 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].
%F A305749 T(n,k) = Sum_{j=1..k} A304972(n,j).
%e A305749 The array begins at T(1,1):
%e A305749 1 1 1 1 1 1 1 1 1 1 1 1 1 ...
%e A305749 1 2 2 2 2 2 2 2 2 2 2 2 2 ...
%e A305749 1 2 3 3 3 3 3 3 3 3 3 3 3 ...
%e A305749 1 4 6 7 7 7 7 7 7 7 7 7 7 ...
%e A305749 1 4 9 11 12 12 12 12 12 12 12 12 12 ...
%e A305749 1 8 18 27 30 31 31 31 31 31 31 31 31 ...
%e A305749 1 8 27 43 55 58 59 59 59 59 59 59 59 ...
%e A305749 1 16 54 107 141 159 163 164 164 164 164 164 164 ...
%e A305749 1 16 81 171 266 312 334 338 339 339 339 339 339 ...
%e A305749 1 32 162 427 688 883 963 993 998 999 999 999 999 ...
%e A305749 1 32 243 683 1313 1774 2069 2169 2204 2209 2210 2210 2210 ...
%e A305749 1 64 486 1707 3407 5103 6119 6634 6789 6834 6840 6841 6841 ...
%e A305749 1 64 729 2731 6532 10368 13524 15080 15790 15975 16026 16032 16033 ...
%e A305749 a(n) are the terms of this array read by antidiagonals.
%e A305749 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.
%t A305749 Ach[n_, k_] := Ach[n,k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] +
%t A305749 Ach[n-2,k-1] + Ach[n-2,k-2]]; (* A304972 *)
%t A305749 Table[Sum[Ach[n, j], {j, 1, k - n + 1}], {k, 1, 15}, {n, 1, k}] // Flatten
%Y A305749 Columns 1-6 are A057427, A016116, A182522, A305750, A305751, and A305752.
%Y A305749 Columns converge to the right to A080107.
%K A305749 nonn,tabl,easy
%O A305749 1,5
%A A305749 _Robert A. Russell_, Jun 09 2018