login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A056329 Number of reversible string structures with n beads using exactly five different colors. 4
0, 0, 0, 0, 1, 9, 76, 542, 3523, 21393, 123680, 690550, 3756151, 20042589, 105394296, 548123982, 2826435403, 14479204833, 73794961960, 374603884910, 1895632969351, 9568915372269, 48208452866816 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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 five 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..23.

FORMULA

a(n) = A056324(n) - A056323(n).

Empirical g.f.: -x^5*(70*x^5 - 102*x^4 + 22*x^3 + 7*x^2 - 4*x + 1) / ((x-1)*(2*x-1)*(2*x+1)*(3*x-1)*(4*x-1)*(5*x-1)*(2*x^2-1)*(5*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=5 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) = (A000481(n) + A304975(n)) / 2 = A000481(n) - A320528(n) = A320528(n) + A304975(n). (End)

EXAMPLE

For a(6)=9, the color patterns are ABCDEA, ABCDBA, ABCCDE, AABCDE, ABACDE, ABCADE, ABCDAE, ABBCDE, and ABCBDE. The first three are achiral. - Robert A. Russell, Oct 14 2018

MATHEMATICA

k=5; Table[(StirlingS2[n, k] + If[EvenQ[n], 3StirlingS2[n/2+2, 5] - 11StirlingS2[n/2+1, 5] + 6StirlingS2[n/2, 5], StirlingS2[(n+5)/2, 5] - 3StirlingS2[(n+3)/2, 5]])/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=5; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)

LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 1, 9, 76, 542, 3523, 21393}, 30] (* Robert A. Russell, Oct 14 2018 *)

CROSSREFS

Column 5 of A284949.

Cf. A056312.

Cf. A000481 (oriented), A320528 (chiral), A304975 (achiral).

Sequence in context: A255931 A319957 A056339 * A190982 A082677 A185818

Adjacent sequences:  A056326 A056327 A056328 * A056330 A056331 A056332

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 14 06:53 EDT 2019. Contains 327995 sequences. (Running on oeis4.)