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%I #21 Mar 13 2024 13:35:14
%S 0,0,0,1,2,5,10,19,38,65,130,211,422,665,1330,2059,4118,6305,12610,
%T 19171,38342,58025,116050,175099,350198,527345,1054690,1586131,
%U 3172262,4766585,9533170,14316139,28632278,42981185,85962370,129009091,258018182,387158345,774316690,1161737179,2323474358
%N Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 3 colors (sets).
%C Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 5, 0, -6).
%F a(n) = [n==0 mod 2] * (2*S2(n/2+1, 3) - 2*S2(n/2, 3)) + [n==1 mod 2] * (S2((n+3)/2, 3) - S2((n+1)/2, 3)) where S2(n,k) is the Stirling subset number A008277(n,k).
%F G.f.: x^3 * (1+2x) / ((1-2x^2) * (1-3x^2)).
%F a(n) = A304972(n,3).
%F a(2m-1) = A140735(m,3).
%F a(2m) = A293181(m,3).
%e For a(5) = 5, the color patterns for both rows and loops are AABCC, ABACA, ABBBC, ABCAB, and ABCBA.
%t Table[If[EvenQ[n], 2 StirlingS2[n/2+1, 3] - 2 StirlingS2[n/2, 3], StirlingS2[(n + 3)/2, 3] - StirlingS2[(n + 1)/2, 3]], {n, 0, 30}]
%t Join[{0}, LinearRecurrence[{0, 5, 0, -6}, {0, 0, 1, 2}, 40]] (* _Robert A. Russell_, Oct 14 2018 *)
%Y Third column of A304972.
%Y Third column of A140735 for odd n.
%Y Third column of A293181 for even n.
%Y Coefficients that determine the first formula and generating function are row 3 of A305008.
%K nonn,easy
%O 0,5
%A _Robert A. Russell_, May 22 2018
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