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 A056327 Number of reversible string structures with n beads using exactly three different colors. 8
 0, 0, 1, 4, 15, 50, 160, 502, 1545, 4730, 14356, 43474, 131145, 395150, 1188580, 3572902, 10732065, 32225810, 96733636, 290322394, 871200825, 2614097750, 7843255300, 23531775502, 70599259185, 211805902490 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 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 three 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 G. C. Greubel, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (6,-6,-24,49,6,-66,36). FORMULA a(n) = A001998(n-1) - A005418(n). G.f.: x^3*(3*x^4 - 8*x^3 + 3*x^2 + 2*x - 1)/((x-1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)). - Colin Barker, Sep 23 2012 From Robert A. Russell, Oct 14 2018: (Start) a(n) = (S2(n,k) + A(n,k))/2, where k=3 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]. a(n) = (A000392(n) + A304973(n)) / 2 = A000392(n) - A320526(n) = A320526(n) + A304973(n). (End) EXAMPLE For a(4)=4, the color patterns are ABCA, ABBC, AABC, and ABAC. The first two are achiral. - Robert A. Russell, Oct 14 2018 MATHEMATICA k=3; Table[(StirlingS2[n, k] + If[EvenQ[n], 2StirlingS2[n/2+1, 3] - 2StirlingS2[n/2, 3], StirlingS2[(n+3)/2, 3] - StirlingS2[(n+1)/2, 3]])/2, {n, 30}] (* Robert A. Russell, Oct 15 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=3; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 30}] (* Robert A. Russell, Oct 15 2018 *) LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 1, 4, 15, 50, 160}, 30] (* Robert A. Russell, Oct 15 2018 *) PROG (PARI) m=40; v=concat([0, 0, 1, 4, 15, 50, 160], vector(m-7)); for(n=8, m, v[n] = 6*v[n-1] -6*v[n-2] -24*v[n-3] +49*v[n-4] +6*v[n-5] -66*v[n-6] +36*v[n-7] ); v \\ G. C. Greubel, Oct 16 2018 (MAGMA) I:=[0, 0, 1, 4, 15, 50, 160]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -24*Self(n-3) +49*Self(n-4) +6*Self(n-5) -66*Self(n-6) +36*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018 CROSSREFS Column 3 of A284949. Cf. A056310. Cf. A000392 (oriented), A320526 (chiral), A304973 (achiral). Sequence in context: A301973 A132308 A026110 * A026328 A014532 A094705 Adjacent sequences:  A056324 A056325 A056326 * A056328 A056329 A056330 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 17 14:47 EDT 2019. Contains 328114 sequences. (Running on oeis4.)