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A056323 Number of reversible string structures with n beads using a maximum of four different colors. 7
1, 1, 2, 4, 11, 31, 107, 379, 1451, 5611, 22187, 87979, 350891, 1400491, 5597867, 22379179, 89500331, 357952171, 1431743147, 5726775979, 22906841771, 91626580651, 366505274027, 1466017950379, 5864067607211 (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 four or fewer nonempty subsets. - Robert A. Russell, Oct 28 2018

There are nonrecursive formulas, generating functions, and computer programs for A124303 and A305750, 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

Table of n, a(n) for n=0..24.

Index entries for linear recurrences with constant coefficients, signature (5,0,-20,16).

FORMULA

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

For n>0, a(n) = (16+(-2)^n+15*2^n+4^n)/48. - Colin Barker, Nov 24 2012

G.f.: (1-4x-3x^2+14x^3-5x^4) / ((1-x)*(1-4x)*(1-4x^2)). [Colin Barker, Nov 24 2012] [Adapted to offset 0 by Robert A. Russell, Nov 09 2018]

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

a(n) = (A124303(n) + A305750(n)) / 2.

a(n) = A124303(n) - A320934(n) = A320934(n) + A305750(n).

a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=4 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). (End)

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=4; Table[Sum[StirlingS2[n, j]+Ach[n, j], {j, 0, k}]/2, {n, 0, 40}] (* Robert A. Russell, Oct 28 2018 *)

LinearRecurrence[{5, 0, -20, 16}, {1, 1, 2, 4, 11}, 40] (* Robert A. Russell, Oct 28 2018 *)

CROSSREFS

Cf. A032121.

Column 4 of A320750.

Cf. A124303 (oriented), A320934 (chiral), A305750 (achiral).

Sequence in context: A190452 A275426 A115625 * A081557 A154603 A063254

Adjacent sequences:  A056320 A056321 A056322 * A056324 A056325 A056326

KEYWORD

nonn,easy

AUTHOR

Marks R. Nester

EXTENSIONS

a(0)=1 prepended by Robert A. Russell, Nov 09 2018

STATUS

approved

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Last modified August 25 05:19 EDT 2019. Contains 326318 sequences. (Running on oeis4.)