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A193981 Number of ways to arrange 3 nonattacking triangular rooks on an nXnXn triangular grid.

%I A193981 #18 Jul 22 2025 12:31:50
%S A193981 0,0,0,2,23,127,468,1352,3310,7190,14260,26330,45885,76237,121688,
%T A193981 187712,281148,410412,585720,819330,1125795,1522235,2028620,2668072,
%U A193981 3467178,4456322,5670028,7147322,8932105,11073545,13626480,16651840,20217080
%N A193981 Number of ways to arrange 3 nonattacking triangular rooks on an nXnXn triangular grid.
%C A193981 Column 3 of A193986
%H A193981 R. H. Hardin, <a href="/A193981/b193981.txt">Table of n, a(n) for n = 1..200</a>
%H A193981 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 A193981 Empirical: a(n) = 6*a(n-1) -14*a(n-2) +14*a(n-3) -14*a(n-5) +14*a(n-6) -6*a(n-7) +a(n-8)
%F A193981 Contribution from _Vaclav Kotesovec_, Aug 31 2012: (Start)
%F A193981 Empirical: G.f.: -x^4*(2 + 11*x + 17*x^2)/((-1+x)^7*(1+x))
%F A193981 Empirical: a(n) = 13*n/24 - 11*n^2/24 - 23*n^3/48 + 9*n^4/16 - 3*n^5/16 + n^6/48 + 1/4*floor(n/2)
%F A193981 (End)
%e A193981 Some solutions for 5X5X5
%e A193981 ......0..........0..........0..........0..........0..........0..........0
%e A193981 .....0.0........0.0........0.0........0.0........0.1........0.0........0.1
%e A193981 ....0.0.1......1.0.0......0.1.0......0.1.0......0.0.0......0.1.0......1.0.0
%e A193981 ...0.1.0.0....0.0.0.1....1.0.0.0....0.0.0.1....1.0.0.0....1.0.0.0....0.0.0.0
%e A193981 ..1.0.0.0.0..0.1.0.0.0..0.0.1.0.0..0.0.1.0.0..0.0.1.0.0..0.0.0.0.1..0.0.0.1.0
%K A193981 nonn
%O A193981 1,4
%A A193981 _R. H. Hardin_ Aug 10 2011