A056323 -id:A056323 - cp's OEIS frontend

A056333 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of four different colors.

1, 1, 3, 9, 30, 102, 378, 1440, 5607, 22155, 87978, 350775, 1400490, 5597487, 22379145, 89498880, 357952170, 1431737433, 5726775978, 22906819575, 91626580269, 366505186047, 1466017950378, 5864067254880

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

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]

Cf. A056315.

Sum mu(d)*A056323(n/d) where d|n.

A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 80, 336, 1344, 5440, 21760, 87296, 349184, 1397760, 5591040, 22368256, 89473024, 357908480, 1431633920, 5726601216, 22906404864, 91625881600, 366503526400, 1466015154176, 5864060616704, 23456246661120, 93824986644480, 375299963355136, 1501199853420544, 6004799480791040

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A124303 and A305750, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Column 4 of A320751.

Cf. A124303 (oriented), A056323 (unoriented), A305750 (achiral).

Table[(4^n - 4^Floor[n/2+1])/48, {n, 40}] (* or *)
LinearRecurrence[{4, 4, -16}, {0, 0, 1}, 40] (* or *)
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,k}]/2,{n,40}]
CoefficientList[Series[x^2/((-1 + 4 x) (-1 + 4 x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *)

a(n) = (A124303(n) - A305750(n))/2.
a(n) = A124303(n) - A056323(n).
a(n) = A056323(n) - A305750(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n).
a(n) = Sum_{j=1..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(2*m) = (16^m - 4*4^m)/48.
a(2*m-1) = (16^m - 4*4^m)/192.
a(n) = (4^n - 4^floor(n/2+1))/48.
G.f.: x^2/((-1 + 4*x)*(-1 + 4*x^2)). - Stefano Spezia, Oct 29 2018
a(n) = 2^n*(2^n - (-1)^n - 3)/48. - Bruno Berselli, Oct 31 2018

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A056333 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of four different colors.

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A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).

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