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A305750 Number of achiral color patterns (set partitions) in a row or cycle of length n with 4 or fewer colors (subsets). 6
1, 1, 2, 3, 7, 11, 27, 43, 107, 171, 427, 683, 1707, 2731, 6827, 10923, 27307, 43691, 109227, 174763, 436907, 699051, 1747627, 2796203, 6990507, 11184811, 27962027, 44739243, 111848107, 178956971, 447392427, 715827883, 1789569707, 2863311531, 7158278827 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABCD are equivalent, as are AABCD and BBCDA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABBCD =  BBCDAA = CDAABB.

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..2000

Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

FORMULA

a(n) = Sum_{j=0..4} 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<2 & n==k].

G.f.: (1 - 3x^2 + x^3) / ((1-x) * (1-4x^2)).

a(2m) = S2(m+4,4) - 8*S2(m+3,4) + 22*S2(m+2,4) - 23*S2(m+1,4) + 6*S2(m,4);

a(2m-1) = S2(m+3,4) - 7*S2(m+2,4) + 16*S2(m+1,4) - 12*S2(m,4), where S2(n,k) is the Stirling subset number A008277.

For m>0, a(2m) = (4 + 5*4^m) / 12.

a(2m-1) = (2 + 4^m) / 6.

a(n) = 2*A056323(n) - A124303(n) = A124303(n) - 2*A320934(n) = A056323(n) - A320934(n).

a(n) = 2*A056354(n) - A056292(n) = A056292(n) - 2*A320744(n) = A056354(n) - A320744(n).

a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n).

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3). - Muniru A Asiru, Oct 28 2018

EXAMPLE

For a(4) = 7, the achiral row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD. The cycle patterns are AAAA, AAAB, AABB, ABAB, AABC, ABAC, and ABCD.

MAPLE

seq(coeff(series((1-3*x^2+x^3)/((1-x)*(1-4*x^2)), x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 28 2018

MATHEMATICA

Table[If[EvenQ[n], StirlingS2[(n+8)/2, 4] - 8 StirlingS2[(n+6)/2, 4] + 22 StirlingS2[(n+4)/2, 4] - 23 StirlingS2[(n+2)/2, 4] + 6 StirlingS2[n/2, 4], StirlingS2[(n+7)/2, 4] - 7 StirlingS2[(n+5)/2, 4] + 16 StirlingS2[(n+3)/2, 4] - 12 StirlingS2[(n+1)/2, 4]], {n, 0, 40}]

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 *)

k=4; Table[Sum[Ach[n, j], {j, 0, k}], {n, 0, 40}]

(* or *)

CoefficientList[Series[(1-3x^2+x^3)/((1-x)(1-4x^2)), {x, 0, 40}], x]

(* or *)

Join[{1}, LinearRecurrence[{1, 4, -4}, {1, 2, 3}, 40]]

(* or *)

Join[{1}, Table[If[EvenQ[n], (4+5 4^(n/2))/12, (2+4^((n+1)/2))/6], {n, 40}]]

PROG

(GAP) a:=[1, 2, 3];; for n in [4..40] do a[n]:=a[n-1]+4*a[n-2]-4*a[n-3]; od; Concatenation([1], a); # Muniru A Asiru, Oct 28 2018

CROSSREFS

Fourth column of A305749.

Cf. A124303 (oriented), A056323 (unoriented), A320934 (chiral), for rows.

Cf. A056292 (oriented), A056354 (unoriented), A320744 (chiral), for cycles.

Sequence in context: A036651 A049454 A095055 * A107857 A107858 A214938

Adjacent sequences:  A305747 A305748 A305749 * A305751 A305752 A305753

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Jun 09 2018

STATUS

approved

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Last modified October 21 19:29 EDT 2019. Contains 328315 sequences. (Running on oeis4.)