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A056310
Number of reversible strings with n beads using exactly three different colors.
5
0, 0, 3, 18, 78, 273, 921, 2916, 9150, 28065, 85773, 259848, 785778, 2367813, 7128201, 21427956, 64382550, 193326105, 580372293, 1741847328, 5227116378, 15684323853, 47059266081, 141189861996
OFFSET
1,3
COMMENTS
A string and its reverse are considered to be equivalent.
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]
FORMULA
a(n) = A032120(n) - 3*A005418(n+1) + 3.
G.f.: -3*x^3*(12*x^4 - 5*x^3 - 4*x^2 + 1)/((x - 1)*(2*x - 1)*(3*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)). [Colin Barker, Jul 07 2012]
EXAMPLE
For n=3, the three rows are ABC, ACB, and BAC, being respectively equivalent to CBA, BCA, and CAB, with which they form chiral pairs. - Robert A. Russell, Sep 25 2018
MAPLE
seq(coeff(series(-3*x^3*(12*x^4-5*x^3-4*x^2+1)/((x-1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)), x, n+1), x, n), n = 1..25); # Muniru A Asiru, Sep 27 2018
MATHEMATICA
k=3; Table[(StirlingS2[i, k]+StirlingS2[Ceiling[i/2], k])k!/2, {i, k, 30}] (* Robert A. Russell, Nov 25 2017 *)
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 3, 18, 78, 273, 921}, 40] (* Vincenzo Librandi, Sep 27 2018 *)
CROSSREFS
Column 3 of A305621.
Equals (A001117 + A056454) / 2 = A001117 - A305623 = A305623 + A056454.
Sequence in context: A043008 A241628 A056319 * A309259 A308853 A309257
KEYWORD
nonn,easy
STATUS
approved