A243211
Triangle T(n, k) = Numbers of 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, 3, 3, 1, 6, 15, 15, 3, 1, 10, 45, 107, 128, 63, 10, 1, 15, 105, 428, 1062, 1566, 1276, 507, 69, 1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 1, 1, 28, 378, 3198, 18591, 77124, 231090, 498097, 759117, 792942, 540361, 222597, 49053
Offset: 1
The triangle begins:
1, 1;
1, 3, 3;
1, 6, 15, 15, 3;
1, 10, 45, 107, 128, 63, 10,
1, 15, 105, 428, 1062, 1566, 1276, 507, 69,
1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 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,
A243207,
A084546,
A234251,
A239567,
A240439,
A194136,
A000217 (column 2),
A050534 (column 3),
A243212 (column 4),
A243213 (column 5),
A243214 (column 6).
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
A237529
Number of ways to choose 4 points in an n X n X n triangular grid so that no 3 of them form a 2 X 2 X 2 subtriangle.
Original entry on oeis.org
6, 156, 1191, 5565, 19620, 57351, 146391, 336951, 714555, 1417515, 2660196, 4763226, 8191911, 13604220, 21909810, 34341666, 52542036, 78664446, 115493685, 166585755, 236429886, 330634821, 456141681, 621465825, 836970225, 1115172981, 1471091706, 1922627616
Offset: 3
- 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).
-
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{6,156,1191,5565,19620,57351,146391,336951,714555},40] (* Harvey P. Dale, Sep 29 2019 *)
-
Vec(-3*x^3*(2*x^6-11*x^5+21*x^4-14*x^3+x^2+34*x+2)/(x-1)^9 + O(x^100)) \\ Colin Barker, Feb 09 2014
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