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

3,1

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).

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)

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 *)

Prog

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

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).

Sequence in context: A028230 A067560 A019553 * A112496 A000483 A162785

Adjacent sequences: A234246 A234247 A234248 * A234250 A234251 A234252

Keyword

nonn,easy

Author

Heinrich Ludwig, Feb 02 2014

Status

approved