A234251
Triangle T(n, k) = Number of ways to choose k points from an n X n X n triangular grid so that no three of them form a 2 X 2 X 2 subtriangle. Triangle T read by rows.
Original entry on oeis.org
1, 1, 1, 3, 3, 1, 6, 15, 16, 6, 1, 10, 45, 111, 156, 120, 42, 2, 1, 15, 105, 439, 1191, 2154, 2583, 1977, 885, 189, 9, 1, 21, 210, 1305, 5565, 17052, 38337, 63576, 77208, 67285, 40512, 15750, 3480, 333, 9, 1, 28, 378, 3240, 19620, 88590, 307362, 833228, 1779219
Offset: 1
Triangle begins
1, 1;
1, 3, 3;
1, 6, 15, 16, 6;
1, 10, 45, 111, 156, 120, 42, 2;
1, 15, 105, 439, 1191, 2154, 2583, 1977, 885, 189, 9;
...
There are no more than T(4, 7) = 2 ways to choose 7 points (X) from a 4 X 4 X 4 grid so that no 3 of them form a 2 X 2 X 2 subtriangle:
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).
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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 *)
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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
A237530
Number of non-equivalent (mod D_3) ways to choose three points in an n X n X n triangular grid so that they do not form a 2 X 2 X 2 subtriangle.
Original entry on oeis.org
0, 4, 22, 82, 231, 566, 1216, 2410, 4428, 7712, 12780, 20392, 31409, 47032, 68594, 97878, 136836, 187998, 254100, 338602, 445213, 578524, 743424, 945860, 1192126, 1489768, 1846734, 2272430, 2776725, 3371170, 4067840, 4880734, 5824442, 6915732, 8172036, 9613236
Offset: 2
- Heinrich Ludwig, Table of n, a(n) for n = 2..1000
- Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1)
-
LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{0,4,22,82,231,566,1216,2410,4428,7712,12780,20392},40] (* Harvey P. Dale, Dec 09 2021 *)
Showing 1-3 of 3 results.
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