OFFSET
1,5
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 four 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
Index entries for linear recurrences with constant coefficients, signature (8, -12, -44, 121, 12, -228, 144).
FORMULA
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
From Robert A. Russell, Oct 14 2018: (Start)
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].
EXAMPLE
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
MATHEMATICA
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 *)
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 = 4; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 1, 6, 37, 183}, 30] (* Robert A. Russell, Oct 14 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved