A234249 Number of ways to choose 4 points in an n X n X n triangular grid.

15, 210, 1365, 5985, 20475, 58905, 148995, 341055, 720720, 1426425, 2672670, 4780230, 8214570, 13633830, 21947850, 34389810, 52602165, 78738660, 115584315, 166695375, 236561325, 330791175, 456326325, 621682425, 837222750, 1115465715, 1471429260, 1923014940

Offset: 3

Author

Heinrich Ludwig, Feb 02 2014

Keywords

• nonn
• easy

Comments

Sequence is column #5 of A084546: a(n) = A084546(n+1, 4).

All elements of the sequence are multiples of 15.

a(n-1) is the number of chiral pairs of colorings of the 8 cubic facets of a tesseract (hypercube) with Schläfli symbol {4,3,3} or of the 8 vertices of a hyperoctahedron with Schläfli symbol {3,3,4}. Both figures are regular 4-D polyhedra and they are mutually dual. Each member of a chiral pair is a reflection, but not a rotation, of the other. - Robert A. Russell, Oct 20 2020

Links

• Heinrich Ludwig, Table of n, a(n) for n = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A084546, A050534 (number of ways to choose 2 points), A093566 (3 points), A231653.

Cf. A337956 (oriented), A337956 (unoriented), A337956 (achiral) colorings, A331356 (hyperoctahedron edges, tesseract faces), A331360 (hyperoctahedron faces, tesseract edges), A337954 (hyperoctahedron facets, tesseract vertices).

Other polychora: A000389 (5-cell), A338950 (24-cell), A338966 (120-cell, 600-cell).

Row 4 of A325006 (orthotope facets, orthoplex vertices).

Programs

Maple

A234249:=n->n*(n + 1)*(n - 1)*(n + 2)*(n - 2)*(n + 3)*(n^2 + n - 4)/384: seq(A234249(n), n=3..40); # Wesley Ivan Hurt, Jan 10 2017

Mathematica

Table[Binomial[Binomial[n,2],4], {n,4,30}] (* Robert A. Russell, Oct 20 2020 *)

PARI

Vec(-15*x^3*(x^2+5*x+1)/(x-1)^9 + O(x^100)) \\ Colin Barker, Feb 02 2014

Formula

a(n) = n*(n + 1)*(n - 1)*(n + 2)*(n - 2)*(n + 3)*(n^2 + n - 4)/384.

a(n) = C(C(n + 1, 2), 4).

G.f.: -15*x^3*(x^2+5*x+1) / (x-1)^9. - Colin Barker, Feb 02 2014

From Robert A. Russell, Oct 20 2020: (Start)

a(n-1) = 15*C(n,4) + 135*C(n,5) + 330*C(n,6) + 315*C(n,7) + 105*C(n,8), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.

a(n-1) = A337956(n) - A337957(n) = (A337956(n) - A337958(n)) / 2 = A337957(n) - A337958(n).

a(n-1) = A325006(4,n). (End)