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 A056328 Number of reversible string structures with n beads using exactly four different colors. 6

%I

%S 0,0,0,1,6,37,183,877,3930,17185,73095,306361,1267266,5198557,

%T 21182343,85910917,347187210,1399451545,5629911015,22616256721,

%U 90754855026,363890126677,1458172596903,5840531635357,23385650196090

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

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

%C Number of set partitions for an unoriented row of n elements using exactly four different elements. An unoriented row is equivalent to its reverse. - _Robert A. Russell_, Oct 14 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]

%F a(n) = A056323(n) - A001998(n-1).

%F Empirical g.f.: -x^4*(3*x^3 + x^2 - 2*x + 1) / ((x-1)*(2*x-1)*(2*x+1)*(3*x-1)*(4*x-1)*(3*x^2-1)). - _Colin Barker_, Nov 25 2012

%F From _Robert A. Russell_, Oct 14 2018: (Start)

%F a(n) = (S2(n,k) + A(n,k))/2, where k=4 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].

%F a(n) = (A000453(n) + A304974(n)) / 2 = A000453(n) - A320527(n) = A320527(n) + A304974(n). (End)

%e For a(5)=6, the color patterns are ABCDA, ABCBD, AABCD, ABACD, ABCAD, and ABBCD. The first two are achiral. - _Robert A. Russell_, Oct 14 2018

%t k=4; Table[(StirlingS2[n,k] + If[EvenQ[n], StirlingS2[n/2+2,4] - StirlingS2[n/2+1,4] - 2StirlingS2[n/2,4], 2StirlingS2[(n+3)/2,4] - 4StirlingS2[(n+1)/2,4]])/2, {n,30}] (* _Robert A. Russell_, Oct 14 2018 *)

%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]]

%t k = 4; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* _Robert A. Russell_, Oct 14 2018 *)

%t LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 1, 6, 37, 183}, 30] (* _Robert A. Russell_, Oct 14 2018 *)

%Y Column 4 of A284949.

%Y Cf. A056311.

%Y Cf. A000453 (oriented), A320527 (chiral), A304974 (achiral).

%K nonn

%O 1,5

%A _Marks R. Nester_

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Last modified October 21 08:47 EDT 2019. Contains 328292 sequences. (Running on oeis4.)