A231653 -id:A231653 - cp's OEIS frontend

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

Heinrich Ludwig, Feb 02 2014

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

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).

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).

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

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

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

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)

A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 4, 2, 1, 1, 3, 10, 25, 41, 48, 41, 25, 10, 3, 1, 1, 4, 22, 87, 244, 526, 870, 1110, 1110, 870, 526, 244, 87, 22, 4, 1, 1, 5, 41, 238, 1029, 3450, 9147, 19524, 34104, 49231, 59038, 59038, 49231, 34104, 19524, 9147, 3450, 1029, 238

Offset: 0

Heinrich Ludwig, Nov 14 2013

Number of orbits under dihedral group D_6 of order 6. - N. J. A. Sloane, Sep 12 2019

			Triangle T(n, k) is irregularly shaped: 0 <= k <= n*(n+1)/2+1. The first row corresponds to n = 1, the first column corresponds to k = 0. Rows are palindromic.
  1,  1;
  1,  1,  1,  1;
  1,  2,  4,  6,  4,  2,  1;
  1,  3, 10, 25, 41, 48, 41, 25, 10,  3,  1;
  ...
There are T(3, 2) = 4 nonisomorphic choices of 2 points (X) in an equilateral triangle grid of side 3:
      X       .       .       X
     . .     X X     . .     X .
    . X .   . . .   X . X   . . .

Heinrich Ludwig, Table of n, a(n) for n = 0..173

Columns k=1..5 are A001399, A227327, A230723, A231653, A231654.

Cf. A054252, A283113, A289709, A326611.

A231654 Number of non-equivalent ways to choose 5 points in an equilateral triangle grid of side n.

Original entry on oeis.org

0, 0, 2, 48, 526, 3450, 16536, 63104, 204202, 580669, 1491096, 3520768, 7754502, 16098425, 31770760, 59998736, 109022244, 191454654, 326158974, 540703008, 874630262, 1383621756, 2144889472, 3263884272, 4882793214, 7190910467, 10437526372, 14947411024

Offset: 1

Heinrich Ludwig, Nov 13 2013

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1).

Cf. A001399, A227327, A230723, A231653, A231655.

Table[If[EvenQ[n], b = 0, b = 375*n^4 - 1170*n^3 + 210*n^2 - 405*n + 1035]; (n^10 + 5*n^9 - 10*n^8 - 70*n^7 + 25*n^6 + 584*n^5 - 420*n^4 - 480*n^3 - 1216*n^2 + 1536*n + b)/23040, {n, 30}] (* T. D. Noe, Nov 14 2013 *)

a(n) = (n^10 + 5*n^9 - 10*n^8 - 70*n^7 + 25*n^6 + 584*n^5 - 420*n^4 - 480*n^3 - 1216*n^2 + 1536*n + B)/23040 where B = 375*n^4 - 1170*n^3 + 210*n^2 - 405*n + 1035 if n odd, and B = 0 if n even.

G.f.: x^3*(x^11 -4*x^10 +14*x^9 -78*x^8 -189*x^7 -902*x^6 -1316*x^5 -1476*x^4 -794*x^3 -258*x^2 -36*x -2) / ((x -1)^11*(x +1)^5). - Colin Barker, Feb 15 2014

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

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A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

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A231654 Number of non-equivalent ways to choose 5 points in an equilateral triangle grid of side n.

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