A305542 Number of chiral pairs of color loops of length n with exactly 3 different colors.

0, 0, 1, 3, 12, 35, 111, 318, 934, 2634, 7503, 21071, 59472, 167229, 472133, 1333263, 3777600, 10721837, 30516447, 87035631, 248820816, 712751271, 2045784183, 5882388956, 16942974060, 48876617790, 141204945463, 408495109005, 1183247473872, 3431451145390, 9962348798055, 28953196894668

Offset: 1

Author

Robert A. Russell, Jun 04 2018

Keywords

• nonn
• easy

Examples

			For a(4)=3, the chiral pairs of color loops are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.
		

Crossrefs

Third column of A305541.

Programs

Mathematica

k=3; Table[(k!/(2n)) DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k] &] - (k!/4) (StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k]), {n, 1, 40}]

PARI

a(n) = my(k=3); -(k!/4)*(stirling(floor((n+1)/2),k,2) + stirling(ceil((n+1)/2),k,2)) + (k!/(2*n))*sumdiv(n, d, eulerphi(d)*stirling(n/d,k,2)); \\ Michel Marcus, Jun 06 2018

Formula

a(n) = -(k!/4)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/(2n))*Sum_{d|n} phi(d)*S2(n/d,k), with k=3 different colors used and where S2(n,k) is the Stirling subset number A008277.

a(n) = (A052823(n) - A056489(n)) / 2.

a(n) = A305541(n,3).

G.f.: -(3/2) * x^4 * (1+x)^2 / Product_{j=1..3} (1-j*x^2) - Sum_{d>0} (phi(d)/(2d)) * (log(1-3x^d) - 3*log(1-2x^d) + 3*log(1-x^d)).