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A304973
Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 3 colors (sets).
9
0, 0, 0, 1, 2, 5, 10, 19, 38, 65, 130, 211, 422, 665, 1330, 2059, 4118, 6305, 12610, 19171, 38342, 58025, 116050, 175099, 350198, 527345, 1054690, 1586131, 3172262, 4766585, 9533170, 14316139, 28632278, 42981185, 85962370, 129009091, 258018182, 387158345, 774316690, 1161737179, 2323474358
OFFSET
0,5
COMMENTS
Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
FORMULA
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).
G.f.: x^3 * (1+2x) / ((1-2x^2) * (1-3x^2)).
a(n) = A304972(n,3).
a(2m-1) = A140735(m,3).
a(2m) = A293181(m,3).
EXAMPLE
For a(5) = 5, the color patterns for both rows and loops are AABCC, ABACA, ABBBC, ABCAB, and ABCBA.
MATHEMATICA
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}]
Join[{0}, LinearRecurrence[{0, 5, 0, -6}, {0, 0, 1, 2}, 40]] (* Robert A. Russell, Oct 14 2018 *)
CROSSREFS
Third column of A304972.
Third column of A140735 for odd n.
Third column of A293181 for even n.
Coefficients that determine the first formula and generating function are row 3 of A305008.
Sequence in context: A132736 A263366 A068035 * A016029 A018327 A285571
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, May 22 2018
STATUS
approved