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A056323 Number of reversible string structures with n beads using a maximum of four different colors. 7

%I

%S 1,1,2,4,11,31,107,379,1451,5611,22187,87979,350891,1400491,5597867,

%T 22379179,89500331,357952171,1431743147,5726775979,22906841771,

%U 91626580651,366505274027,1466017950379,5864067607211

%N Number of reversible string structures with n beads using a maximum of four different colors.

%C 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.

%C Number of set partitions of an unoriented row of n elements with four or fewer nonempty subsets. - _Robert A. Russell_, Oct 28 2018

%C 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

%D 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]

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,0,-20,16).

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

%F For n>0, a(n) = (16+(-2)^n+15*2^n+4^n)/48. - _Colin Barker_, Nov 24 2012

%F 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]

%F From _Robert A. Russell_, Oct 28 2018: (Start)

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

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

%F 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)).

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

%e 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.

%t 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 *)

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

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

%Y Cf. A032121.

%Y Column 4 of A320750.

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

%K nonn,easy

%O 0,3

%A _Marks R. Nester_

%E a(0)=1 prepended by _Robert A. Russell_, Nov 09 2018

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Last modified September 18 09:56 EDT 2019. Contains 327169 sequences. (Running on oeis4.)