A056327 Number of reversible string structures with n beads using exactly three different colors.

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

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: A132308 A372015 A026110 * A026328 A014532 A094705

Adjacent sequences: A056324 A056325 A056326 * A056328 A056329 A056330

Keyword

nonn,easy

Author

Marks R. Nester

Status

approved