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).
A243142
Number of inequivalent (mod D_3) ways to place 3 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, 3, 19, 75, 218, 542, 1178, 2350, 4340, 7585, 12605, 20153, 31094, 46620, 68068, 97212, 136008, 186975, 252855, 337095, 443410, 576378, 740894, 942890, 1188668, 1485757, 1842113, 2267125, 2770670, 3364280, 4060040, 4871928, 5814544, 6904635, 8159643, 9599427
Offset: 2
- Heinrich Ludwig, Table of n, a(n) for n = 2..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-3,-8,14,0,-14,8,3,-4,1).
-
Drop[CoefficientList[Series[x^3*(2*x^5-5*x^4+x^3-8*x^2-7*x-3) / ((x-1)^7*(x+1)^3), {x, 0, 40}], x],2] (* Vaclav Kotesovec, May 31 2014 after Colin Barker *)
-
concat(0, Vec(x^3*(2*x^5-5*x^4+x^3-8*x^2-7*x-3)/((x-1)^7*(x+1)^3) + O(x^100))) \\ Colin Barker, May 30 2014
A243143
Number of inequivalent (mod D_3) ways to place 4 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
1, 22, 170, 816, 2947, 8765, 22703, 52823, 113042, 225817, 426299, 766905, 1324282, 2206478, 3563770, 5599258, 8584775, 12875840, 18934040, 27347390, 38860741, 54402707, 75125825, 102441321, 138070912, 184090795, 242997153, 317760863, 411908932, 529591532, 675681764
Offset: 3
- Heinrich Ludwig, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (5,-7,-5,23,-19,-7,27,-27,7,19,-23,5,7,-5,1).
-
Drop[CoefficientList[Series[-x^3*(3*x^10 - 10*x^9 + 19*x^8 - 13*x^7 + 102*x^6 + 105*x^5 + 144*x^4 + 125*x^3 + 67*x^2 + 17*x + 1) / ((x-1)^9*(x+1)^4*(x^2+1)), {x, 0, 40}], x],3] (* Vaclav Kotesovec, May 31 2014 after Colin Barker *)
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
Showing 1-4 of 4 results.
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