A320524 Number of chiral pairs of a row of n colors using 6 or fewer colors.

0, 15, 90, 630, 3780, 23220, 139320, 839160, 5034960, 30229200, 181375200, 1088367840, 6530207040, 39181942080, 235091652480, 1410554113920, 8463324683520, 50779973295360, 304679839772160, 1828079189798400, 10968475138790400, 65810851739735040, 394865110438410240, 2369190668072417280, 14215144008434503680, 85290864083258757120

Offset

1,2

Comments

A chiral row is different from its reverse.

Links

Table of n, a(n) for n=1..26.

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

Formula

a(n) = (k^n - k^ceiling(n/2)) / 2, where k=6 is maximum number of colors.

G.f.: k*x^2*(k-1) / (2*(1-k*x)*(1-k*x^2)), where k=6.

a(n) = (A000400(n) - A056452(n)) / 2 = A000400(n) - A056308(n) = A056308(n) - A056452(n).

Example

For a(2)=15, the chiral pairs are the fifteen combinations of six colors taken two at a time, e.g., AB-BA.

Mathematica

k = 6; Table[(k^n - k^Ceiling[n/2])/2, {n, 1, 30}]
LinearRecurrence[{6, 6, -36}, {0, 15, 90}, 30]

Prog

(Magma) [(6^n - 6^Ceiling(n / 2)) / 2: n in [1..25]]; // Vincenzo Librandi, Oct 15 2018
(PARI) m=40; v=concat([0, 15, 90], vector(m-3)); for(n=4, m, v[n] = 6*v[n-1] +6*v[n-2] -36*v[n-3]); v \\ G. C. Greubel, Oct 17 2018

Crossrefs

Column 6 of A293500.

Cf. A000400 (oriented), A056308 (unoriented), A056452 (achiral).

Sequence in context: A292055 A263628 A263629 * A218408 A048630 A326148

Adjacent sequences: A320521 A320522 A320523 * A320525 A320526 A320527

Keyword

easy,nonn

Author

Robert A. Russell, Oct 14 2018

Status

approved