A056326 Number of reversible string structures with n beads using exactly two different colors.

0, 1, 2, 5, 9, 19, 35, 71, 135, 271, 527, 1055, 2079, 4159, 8255, 16511, 32895, 65791, 131327, 262655, 524799, 1049599, 2098175, 4196351, 8390655, 16781311, 33558527, 67117055, 134225919, 268451839, 536887295, 1073774591, 2147516415, 4295032831, 8590000127

Offset: 1

Author

Marks R. Nester

Keywords

• nonn
• easy

Comments

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

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

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3, 0, -6, 4).

Crossrefs

Column k=2 of A284949 and of A291883.

Cf. A056309.

Programs

Mathematica

Table[(StirlingS2[n,2] + StirlingS2[Floor[n/2]+1,2])/2, {n,1,30}] (* Robert A. Russell, Jan 29 2018 *)
LinearRecurrence[{3, 0, -6, 4}, {0, 1, 2, 5}, 35] (* or *)
Rest@ CoefficientList[Series[x^2*(x^2 + x - 1)/((x - 1) (2 x - 1) (2 x^2 - 1)), {x, 0, 35}], x] (* Michael De Vlieger, Jan 31 2018 *)

Formula

a(n) = A005418(n) - 1.

From Colin Barker, Nov 25 2012: (Start)

a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).

G.f.: x^2*(x^2+x-1)/((x-1)*(2*x-1)*(2*x^2-1)). (End)