A243207
Triangle T(n, k) = Numbers of inequivalent (mod D_3) ways to place k 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. Triangle read by rows.
Original entry on oeis.org
1, 1, 1, 2, 4, 3, 1, 3, 10, 20, 25, 11, 3, 4, 22, 77, 186, 266, 221, 86, 14, 5, 41, 223, 881, 2344, 4238, 4885, 3451, 1296, 220, 7, 1, 7, 72, 552, 3146, 12907, 38640, 83107, 126701, 132236, 90214, 37128, 8235, 775, 24, 8, 116, 1196, 9264, 53307, 232861, 773930
Offset: 1
The triangle begins:
1;
1, 1;
2, 4, 3, 1;
3, 10, 20, 25, 11, 3;
4, 22, 77, 186, 266, 221, 86, 14;
5, 41, 223, 881, 2344, 4238, 4885, 3451, 1296, 220, 7, 1;
...
There is T(6, 12) = 1 way to place 12 points (x) on the grid obeying the rule in the definition of the sequence:
.
x x
x . x
x . . x
x . . . x
. x x x x .
Cf.
A227308,
A243211,
A239572,
A234247,
A231655,
A243141,
A001399 (column 1),
A227327 (column 2),
A243208 (column 3),
A243209 (column 4),
A243210 (column 5).
A231653
Number of non-equivalent ways to choose 4 points in an equilateral triangle grid of side n.
Original entry on oeis.org
0, 0, 4, 41, 244, 1029, 3485, 9926, 25030, 57126, 120570, 238330, 446344, 797825, 1370684, 2274259, 3660612, 5734776, 8771181, 13127940, 19270240, 27789713, 39435814, 55142010, 76066910, 103627784, 139554142, 185929971, 245260890, 320527585, 415268815
Offset: 1
For n = 3 there are the following a(3) = 4 choices of 4 points (=X) (rotations and reflections ignored):
X . X X
X X X X X X . .
. X . X . X . . X X X X
- Heinrich Ludwig, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).
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 *)
Showing 1-3 of 3 results.
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