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Number of ways to arrange 4 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.
1

%I #13 May 05 2018 08:06:30

%S 0,0,6,114,879,4284,15729,47565,124803,293733,634293,1277133,2426424,

%T 4389567,7615062,12739902,20647962,32540958,50023656,75205116,

%U 110817861,160356966,228241167,319998195,442476645,604086795,815072895,1087819551

%N Number of ways to arrange 4 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

%C Column 4 of A194480.

%H R. H. Hardin, <a href="/A194476/b194476.txt">Table of n, a(n) for n = 1..112</a>

%F Empirical: a(n) = (1/384)*n^8 + (1/96)*n^7 - (5/64)*n^6 + (13/240)*n^5 + (27/128)*n^4 - (23/96)*n^3 - (13/96)*n^2 + (7/40)*n.

%F Empirical g.f.: x^3*(2 + 20*x + 23*x^2 - 9*x^3 - x^4) / (1 - x)^9. - _Colin Barker_, May 05 2018

%e All solutions for 3 X 3 X 3:

%e ....0......1......0......1......1......0

%e ...1.1....0.1....1.1....1.1....1.0....1.1

%e ..0.1.1..1.1.0..1.1.0..0.1.0..0.1.1..1.0.1

%Y Cf. A194480

%K nonn

%O 1,3

%A _R. H. Hardin_, Aug 26 2011