A305623 Number of chiral pairs of rows of n colors with exactly 3 different colors.

0, 0, 3, 18, 72, 267, 885, 2880, 9000, 27915, 85233, 259308, 783972, 2366007, 7122405, 21422160, 64364400, 193307955, 580316313, 1741791348, 5226945372, 15684152847, 47058746925, 141189342840, 423593188200, 1270831465995, 3812595048993, 11437991207388, 34314376250772, 102943948309287, 308833455491445, 926503630549920, 2779517334002400, 8338565015656035, 25015720816575273, 75047214375967428

Offset

1,3

Comments

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Links

Table of n, a(n) for n=1..36.

Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=3 colors used and where S2(n,k) is the Stirling subset number A008277.

a(n) = (A001117(n) - A056454(n)) / 2.

a(n) = A001117(n) - A056310(n) = A056310(n) - A056454(n).

G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=3 colors used.

G.f.: 3*x^3*(5*x^2-x-1)/(-36*x^6+30*x^5+24*x^4-25*x^3-x^2+5*x-1). - Simon Plouffe, Jun 20 2018

Example

For a(3) = 3, the chiral pairs are ABC-CBA, ACB-BCA, and BAC-CAB.

Mathematica

k=3; Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 40}]

Prog

(PARI) a(n) = 3*(stirling(n, 3, 2)-stirling(ceil(n/2), 3, 2)); \\ Altug Alkan, Sep 26 2018

Crossrefs

Third column of A305622.

A056454(n) is number of achiral rows of n colors with exactly 3 different colors.

Cf. A001117, A056310.

Sequence in context: A114633 A135070 A073961 * A280804 A152897 A059393

Adjacent sequences: A305620 A305621 A305622 * A305624 A305625 A305626

Keyword

nonn,easy

Author

Robert A. Russell, Jun 06 2018

Status

approved