A337406 Number of chiral pairs of colorings of the edges of a cube (or regular octahedron) using n or fewer colors.

0, 74, 10704, 345640, 5062600, 45246810, 288005144, 1430618784, 5881281480, 20827126650, 65370603320, 185725346664, 485325996064, 1181031257770, 2702889008400, 5863794289280, 12137528310384, 24099966466794

Offset: 1

Author

Robert A. Russell, Aug 26 2020

Keywords

• nonn
• easy

Comments

Each member of a chiral pair is a reflection, but not a rotation, of the other. Both the cube and the octahedron have 12 edges.

Links

• Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Crossrefs

Cf. A060530 (oriented), A199406 (unoriented), A331351 (achiral).

Row 3 of A337409 (orthotope edge colorings) and A337413 (orthoplex edge colorings).

Programs

Mathematica

Table[(n-1)n^2(n+1)(n^8+n^6-2n^4+8)/48, {n,20}]
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,74,10704,345640,5062600,45246810,288005144,1430618784,5881281480,20827126650,65370603320,185725346664,485325996064},20] (* Harvey P. Dale, Jul 11 2025 *)

Formula

a(n) = (n-1) * n^2 * (n+1) * (n^8 + n^6 - 2n^4 + 8) / 48.

a(n) = 74*C(n,2) + 10482*C(n,3) + 303268*C(n,4) + 3440700*C(n,5) + 19842840*C(n,6) + 65867760*C(n,7) + 133580160*C(n,8) + 168399000*C(n,9) + 128898000*C(n,10) + 54885600*C(n,11) + 9979200*C(n,12), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = (A060530(n) - A331351(n)) / 2 = A060530(n) - A199406(n) = A199406(n) - A331351(n).

G.f.: 2 * (37*x^2 + 4871*x^3 + 106130*x^4 + 691514*x^5 + 1692248*x^6 + 1692248*x^7 + 691514*x^8 + 106130*x^9 + 4871*x^10 + 37*x^11) / (1-x)^13.