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A056324 Number of reversible string structures with n beads using a maximum of five different colors. 9
1, 1, 2, 4, 11, 32, 116, 455, 1993, 9134, 43580, 211659, 1041441, 5156642, 25640456, 127773475, 637624313, 3184387574, 15910947980, 79521737939, 397510726681, 1987259550002, 9935420646296, 49674470817195, 248364482308833, 1241798790172214 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure. Thus aabc, cbaa and bbac are all considered to be identical.

Number of set partitions of an unoriented row of n elements with five or fewer nonempty subsets. - Robert A. Russell, Oct 28 2018

There are nonrecursive formulas, generating functions, and computer programs for A056272 and A305751, which can be used in conjunction with the formula. - Robert A. Russell, Oct 28 2018

REFERENCES

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..1000

FORMULA

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

G.f.: (1-10x+25x^2+32x^3-196x^4+149x^5+225x^6-321x^7+85x^8)/((1-x)*(1-2x)*(1-3x)*(1-5x)*(1-2x^2)*(1-5x^2)). - Colin Barker, Nov 24 2012 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]

From Robert A. Russell, Oct 28 2018: (Start)

a(n) = (A056272(n) + A305751(n)) / 2.

a(n) = A056272(n) - A320935(n) = A320935(n) + A305751(n).

a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=5 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).

a(n) = A000007(n) + A057427(n) + A056326(n) + A056327(n) + A056328(n) + A056329(n). (End)

For n>8, a(n) = 11*a(n-1) - 34*a(n-2) - 16*a(n-3) + 247*a(n-4) - 317*a(n-5) - 200*a(n-6) + 610*a(n-7) - 300*a(n-8). - Muniru A Asiru, Oct 30 2018

EXAMPLE

For a(4)=11, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, ABBC, and ABCD.  The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

MATHEMATICA

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=5; Table[Sum[StirlingS2[n, j]+Ach[n, j], {j, 0, k}]/2, {n, 0, 40}]  (* Robert A. Russell, Oct 28 2018 *)

LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {1, 1, 2, 4, 11, 32, 116, 455, 1993}, 40] (* Robert A. Russell, Oct 28 2018 *)

CROSSREFS

Cf. A032122.

Column 5 of A320750.

Cf. A056272 (oriented), A320935 (chiral), A305751 (achiral).

Sequence in context: A318644 A113774 A124504 * A056325 A103293 A123418

Adjacent sequences:  A056321 A056322 A056323 * A056325 A056326 A056327

KEYWORD

nonn

AUTHOR

Marks R. Nester

EXTENSIONS

Terms added by Robert A. Russell, Oct 30 2018

a(0)=1 prepended by Robert A. Russell, Nov 07 2018

STATUS

approved

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)