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A193982 Number of ways to arrange 4 nonattacking triangular rooks on an nXnXn triangular grid.

%I A193982 #17 Jul 22 2025 12:31:57
%S A193982 0,0,0,0,0,18,233,1449,6213,20993,59943,150903,344323,726033,1434678,
%T A193982 2685046,4798206,8240022,13669026,21995586,34453386,52685556,78846471,
%U A193982 115721991,166869131,236778399,331059729,456655745,622083189,837706779
%N A193982 Number of ways to arrange 4 nonattacking triangular rooks on an nXnXn triangular grid.
%C A193982 Column 4 of A193986
%H A193982 R. H. Hardin, <a href="/A193982/b193982.txt">Table of n, a(n) for n = 1..89</a>
%H A193982 Christopher R. H. Hanusa, Thomas Zaslavsky, <a href="https://arxiv.org/abs/1906.08981">A q-queens problem. VII. Combinatorial types of nonattacking chess riders</a>, arXiv:1906.08981 [math.CO], 2019.
%F A193982 Empirical: a(n) = 6*a(n-1) -12*a(n-2) +2*a(n-3) +27*a(n-4) -36*a(n-5) +36*a(n-7) -27*a(n-8) -2*a(n-9) +12*a(n-10) -6*a(n-11) +a(n-12)
%F A193982 Contribution from _Vaclav Kotesovec_, Aug 31 2012: (Start)
%F A193982 Empirical: G.f.: -x^6*(18 + 125*x + 267*x^2 + 279*x^3 + 151*x^4)/((-1+x)^9*(1+x)^3)
%F A193982 Empirical: a(n) = 87*n/40 - 57*n^2/32 - 253*n^3/96 + 1385*n^4/384 - 139*n^5/80 + 27*n^6/64 - 5*n^7/96 + n^8/384 + (3 - 11*n/8 + n^2/8)*floor(n/2)
%F A193982 (End)
%e A193982 Some solutions for 6X6X6
%e A193982 .......0............0............0............0............0............0
%e A193982 ......0.0..........0.0..........1.0..........0.0..........0.1..........0.0
%e A193982 .....0.0.1........1.0.0........0.0.0........0.1.0........1.0.0........1.0.0
%e A193982 ....0.1.0.0......0.0.0.1......0.0.0.1......0.0.0.1......0.0.0.0......0.0.1.0
%e A193982 ...1.0.0.0.0....0.1.0.0.0....0.0.1.0.0....1.0.0.0.0....0.0.0.1.0....0.1.0.0.0
%e A193982 ..0.0.0.0.1.0..0.0.0.0.1.0..0.1.0.0.0.0..0.0.1.0.0.0..0.0.1.0.0.0..0.0.0.0.0.1
%K A193982 nonn
%O A193982 1,6
%A A193982 _R. H. Hardin_ Aug 10 2011