This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056330 #19 Mar 03 2024 11:28:05
%S A056330 0,0,0,0,0,1,12,142,1346,11511,89974,662674,4662574,31724735,
%T A056330 210361046,1367510326,8752976610,55343947975,346541488998,
%U A056330 2153041587538,13292844257198,81652683550119,499484958151630
%N A056330 Number of reversible string structures with n beads using exactly six different colors.
%C A056330 A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.
%C A056330 Number of set partitions for an unoriented row of n elements using exactly six different elements. An unoriented row is equivalent to its reverse. - _Robert A. Russell_, Oct 14 2018
%D A056330 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 A056330 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600).
%F A056330 a(n) = A056325(n) - A056324(n).
%F A056330 From _Robert A. Russell_, Oct 14 2018: (Start)
%F A056330 a(n) = (S2(n,k) + A(n,k))/2, where k=6 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 A056330 G.f.: (x^6 / Product_{k=1..6} (1 - k*x) + x^6 (1+x) (1-4x^2) (1+2x-x^2-4x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.
%F A056330 a(n) = (A000770(n) + A304976(n)) / 2 = A000770(n) - A320529(n) = A320529(n) + A304976(n). (End)
%e A056330 For a(7)=12, the color patterns are ABCDEFA, ABCDEBF, ABCDCEF, AABCDEF, ABACDEF, ABCADEF, ABCDAEF, ABBCDEF, ABCBDEF, ABCDBEF, and ABCCDEF. The first three are achiral. - _Robert A. Russell_, Oct 14 2018
%t A056330 k=6; Table[(StirlingS2[n,k] + If[EvenQ[n], StirlingS2[n/2+3,6] - 3StirlingS2[n/2+2,6] - 8StirlingS2[n/2+1,6] + 16StirlingS2[n/2,6], 3StirlingS2[(n+5)/2,6] - 17StirlingS2[(n+3)/2,6] + 20StirlingS2[(n+1)/2,6]])/2, {n,30}] (* _Robert A. Russell_, Oct 14 2018 *)
%t A056330 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 A056330 k = 6; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* _Robert A. Russell_, Oct 14 2018 *)
%t A056330 LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 1, 12, 142, 1346, 11511, 89974, 662674, 4662574, 31724735}, 40] (* _Robert A. Russell_, Oct 14 2018 *)
%Y A056330 Column 6 of A284949.
%Y A056330 Cf. A056313.
%Y A056330 Cf. A000770 (oriented), A320529 (chiral), A304976 (achiral).
%K A056330 nonn,easy
%O A056330 1,7
%A A056330 _Marks R. Nester_