A056310 Number of reversible strings with n beads using exactly three different colors.

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]

Links

Vincenzo Librandi, 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) = 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

Cf. A032120, A005418.

Column 3 of A305621.

Equals (A001117 + A056454) / 2 = A001117 - A305623 = A305623 + A056454.

Sequence in context: A043008 A241628 A056319 * A309259 A308853 A309257

Adjacent sequences: A056307 A056308 A056309 * A056311 A056312 A056313

Keyword

nonn,easy

Author

Marks R. Nester

Status

approved