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A056328 Number of reversible string structures with n beads using exactly four different colors. 5
0, 0, 0, 1, 6, 37, 183, 877, 3930, 17185, 73095, 306361, 1267266, 5198557, 21182343, 85910917, 347187210, 1399451545, 5629911015, 22616256721, 90754855026, 363890126677, 1458172596903, 5840531635357, 23385650196090 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..25.

FORMULA

a(n) = A056323(n) - A001998(n-1).

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].

a(n) = (A000453(n) + A304974(n)) / 2 = A000453(n) - A320527(n) = A320527(n) + A304974(n). (End)

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

Column 4 of A284949.

Cf. A056311.

Cf. A000453 (oriented), A320527 (chiral), A304974 (achiral).

Sequence in context: A129552 A293800 A056338 * A156185 A057418 A001419

Adjacent sequences:  A056325 A056326 A056327 * A056329 A056330 A056331

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

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Last modified September 18 12:45 EDT 2019. Contains 327170 sequences. (Running on oeis4.)