A032086 Number of reversible strings with n beads of 3 colors. If more than 1 bead, not palindromic.

3, 3, 9, 36, 108, 351, 1053, 3240, 9720, 29403, 88209, 265356, 796068, 2390391, 7171173, 21520080, 64560240, 193700403, 581101209, 1743362676, 5230088028, 15690441231, 47071323693, 141214502520, 423643507560

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

C. G. Bower, Transforms (2)

Index entries for linear recurrences with constant coefficients, signature (3, 3, -9).

Formula

"BHK" (reversible, identity, unlabeled) transform of 3, 0, 0, 0, ...

Conjectures from Colin Barker, Apr 02 2012: (Start)

a(n) = 3*a(n-1) + 3*a(n-2) - 9*a(n-3) for n > 4.

G.f.: 3*x*(1 - 2*x - 3*x^2 + 9*x^3)/((1 - 3*x)*(1 - 3*x^2)).

(End)

Conjectures from Colin Barker, Mar 09 2017: (Start)

a(n) = (2*3^n - 2*3^(n/2)) / 4 for n > 2 and even.

a(n) = (2*3^n - 2*3^((n+1)/2)) / 4 for n > 2 and odd.

(End)

The above conjectures are true: The second set follows from the definition and the first set can be derived from that. - Andrew Howroyd, Oct 10 2017

a(n) = (3^n - 3^(ceiling(n/2)) / 2 = (A000244(n) - A056449(n)) / 2 for n>1. - Robert A. Russell and Danny Rorabaugh, Jun 22 2018

Mathematica

Join[{3}, LinearRecurrence[{3, 3, -9}, {3, 9, 36}, 24]] (* Jean-François Alcover, Oct 11 2017 *)

Prog

(PARI) a(n) = if(n<2, [3][n], (3^n - 3^(ceil(n/2)))/2); \\ Andrew Howroyd, Oct 10 2017

Crossrefs

Column 3 of A293500 for n>1.

Cf. A032120.

Sequence in context: A264412 A202889 A257620 * A241278 A100239 A245023

Adjacent sequences: A032083 A032084 A032085 * A032087 A032088 A032089

Keyword

nonn

Author

Christian G. Bower

Status

approved