A304973 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 3 colors (sets).

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.

Links

Table of n, a(n) for n=0..40.

Index entries for linear recurrences with constant coefficients, signature (0, 5, 0, -6).

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

Adjacent sequences: A304970 A304971 A304972 * A304974 A304975 A304976

Keyword

nonn,easy

Author

Robert A. Russell, May 22 2018

Status

approved