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

Author

Robert A. Russell, Oct 14 2018

Keywords

• easy
• nonn

Comments

A chiral row is different from its reverse.

Examples

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

Links

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

Crossrefs

Column 6 of A293500.

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

Programs

Magma

[(6^n - 6^Ceiling(n / 2)) / 2: n in [1..25]]; // Vincenzo Librandi, Oct 15 2018

Mathematica

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

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

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).