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

1,4

Links

Table of n, a(n) for n=1..32.

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)).

Example

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

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}]

Prog

(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

Crossrefs

Third column of A305541.

Sequence in context: A260006 A303862 A320346 * A232899 A009787 A135190

Adjacent sequences: A305539 A305540 A305541 * A305543 A305544 A305545

Keyword

nonn,easy

Author

Robert A. Russell, Jun 04 2018

Status

approved