A056308 Number of reversible strings with n beads using a maximum of six different colors.

1, 6, 21, 126, 666, 3996, 23436, 140616, 840456, 5042736, 30236976, 181421856, 1088414496, 6530486976, 39182222016, 235093332096, 1410555793536, 8463334761216, 50779983373056, 304679900238336, 1828079250264576, 10968475501587456, 65810852102532096

Offset

0,2

Comments

A string and its reverse are considered to be equivalent. Thus aabc and cbaa are considered to be identical, but abca is a different string.

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

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,6,-36).

Formula

a(n) = (6^n + 6^floor((n+1)/2))/2.

G.f.: (1-21*x^2) / ((1-6*x)*(1-6*x^2)). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]

a(n) = 6*a(n-1) + 6*a(n-2) - 36*a(n-3) for n > 3. - Colin Barker, Mar 20 2017

a(n) = (A000400(n) + A056452(n)) / 2. - Robert A. Russell, Jun 19 2018

a(n) = 6^(n + floor((n-1)/2)) * a(1-n) for all n in Z. - Michael Somos, Jul 10 2018

Example

For a(2)=21, the six achiral strings are AA, BB, CC, DD, EE, and FF; the 15 (equivalent) chiral pairs are AB-BA, AC-CA, AD-DA, AE-EA, AF-FA, BC-CB, BD-DB, BE-EB, BF-FB, CD-DC, CE-EC, CF-FC, DE-ED, DF-FD, and EF-FE.

Mathematica

k=6; Table[(k^n+k^Ceiling[n/2])/2, {n, 0, 30}] (* Robert A. Russell, Nov 25 2017 *)
a[ n_] := (6^n + 6^Quotient[n + 1, 2]) / 2; (* Michael Somos, Jul 10 2018 *)
LinearRecurrence[{6, 6, -36}, {1, 6, 21}, 31] (* Robert A. Russell, Nov 10 2018 *)

Prog

(PARI) Vec((1-21*x^2) / ((1 - 6*x)*(1 - 6*x^2)) + O(x^30)) \\ Colin Barker, Mar 20 2017 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]
{PARI) {a(n) = (6^n + 6^((n+1)\2)) / 2}; \\ Michael Somos, Jul 10 2018
(Magma) I:=[1, 6, 21]; [n le 3 select I[n] else 6*Self(n-1) +6*Self(n-2) - 36*Self(n-3): n in [1..30]]; // G. C. Greubel, Nov 10 2018

Crossrefs

Column 6 of A277504.

Cf. A000400 (oriented), A320524 (chiral), A056452 (achiral).

Sequence in context: A251593 A012840 A013320 * A081077 A093775 A318103

Adjacent sequences: A056305 A056306 A056307 * A056309 A056310 A056311

Keyword

nonn,easy

Author

Marks R. Nester

Extensions

a(0)=1 prepended by Robert A. Russell, Nov 10 2018

Status

approved