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 A056330 Number of reversible string structures with n beads using exactly six different colors. 3
 0, 0, 0, 0, 0, 1, 12, 142, 1346, 11511, 89974, 662674, 4662574, 31724735, 210361046, 1367510326, 8752976610, 55343947975, 346541488998, 2153041587538, 13292844257198, 81652683550119, 499484958151630 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure. 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 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 FORMULA a(n) = A056325(n) - A056324(n). From Robert A. Russell, Oct 14 2018: (Start) 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]. 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. a(n) = (A000770(n) + A304976(n)) / 2 = A000770(n) - A320529(n) = A320529(n) + A304976(n). (End) EXAMPLE 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 MATHEMATICA 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 *) 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]] k = 6; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *) 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 *) CROSSREFS Column 6 of A284949. Cf. A056313. Cf. A000770 (oriented), A320529 (chiral), A304976 (achiral). Sequence in context: A266177 A135326 A056340 * A158516 A163448 A219307 Adjacent sequences:  A056327 A056328 A056329 * A056331 A056332 A056333 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified May 12 20:38 EDT 2021. Contains 343829 sequences. (Running on oeis4.)