A243210
Number of inequivalent (mod D_3) ways to place 5 points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.
Original entry on oeis.org
0, 11, 266, 2344, 12907, 53307, 180876, 530654, 1391647, 3335627, 7426885, 15544434, 30867669, 58574800, 106838511, 188190111, 321383808, 533857914
Offset: 3
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, 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).
-
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 *)
A234247
Triangle T(n,k) read by rows: Number of non-equivalent ways (mod D_3) to choose k points from an nXnXn triangular grid so that no three of them form a 2X2X2 subtriangle.
Original entry on oeis.org
1, 1, 1, 2, 4, 4, 2, 3, 10, 22, 31, 22, 10, 1, 4, 22, 82, 212, 374, 450, 342, 156, 36, 2, 5, 41, 231, 955, 2880, 6459, 10660, 12948, 11274, 6802, 2645, 595, 57, 2, 7, 72, 566, 3335, 14883, 51470, 139224, 297048, 500147, 661796, 681101, 536322, 314753, 132490
Offset: 1
Triangle begins
1;
1, 1;
2, 4, 4, 2;
3, 10, 22, 31, 22, 10, 1;
4, 22, 82, 212, 374, 450, 342, 156, 36, 2;
5, 41, 231, 955, 2880, 6459, 10660, 12948, 11274, 6802, 2645, 595, 57, 2;
...
There are exactly T(5, 10) = 2 non-equivalent ways to choose 10 points (X) from a triangular grid of side 5 avoiding that any three of them form a subtriangle of side 2.
. X
X X . X
X . X X . X
. X X . . X X .
X X . X X X X . X X
A243144
Number of inequivalent (mod D_3) ways to place 5 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.
Original entry on oeis.org
0, 7, 204, 1891, 10846, 46068, 159830, 477033, 1268614, 3075291, 6911894, 14580293, 29145928, 55620816, 101945063, 180327134, 309087474
Offset: 3
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