cp's OEIS Frontend

A194494 Number of ways to arrange 4 nonattacking queens on the lower triangle of an n X n board.

%I A194494 #11 Jul 22 2025 12:33:25
%S A194494 0,0,0,0,0,8,118,802,3708,13280,39734,104000,244948,530632,1072776,
%T A194494 2048056,3723314,6492342,10915254,17777372,28147380,43465356,65624634,
%U A194494 97098802,141050688,201509798,283514112,393348562,538725268,729098516
%N A194494 Number of ways to arrange 4 nonattacking queens on the lower triangle of an n X n board.
%C A194494 Column 4 of A194498
%H A194494 R. H. Hardin, <a href="/A194494/b194494.txt">Table of n, a(n) for n = 1..81</a>
%F A194494 Empirical: a(n) = 4*a(n-2) +3*a(n-3) -5*a(n-4) -11*a(n-5) -3*a(n-6) +11*a(n-7) +14*a(n-8) +6*a(n-9) -7*a(n-10) -16*a(n-11) -14*a(n-12) +14*a(n-14) +16*a(n-15) +7*a(n-16) -6*a(n-17) -14*a(n-18) -11*a(n-19) +3*a(n-20) +11*a(n-21) +5*a(n-22) -3*a(n-23) -4*a(n-24) +a(n-26), [_R. H. Hardin_, Aug 26 2011]
%F A194494 G.f.: -2*x^6*(287*x^18 + 1545*x^17 + 4929*x^16 + 11689*x^15 + 22673*x^14 + 36995*x^13 + 51875*x^12 + 63203*x^11 + 67465*x^10 + 63168*x^9 + 51807*x^8 + 36900*x^7 + 22544*x^6 + 11587*x^5 + 4879*x^4 + 1606*x^3 + 385*x^2 + 59*x + 4)/((x-1)^9*(x+1)^5*(x^2+1)*(x^2+x+1)^3*(x^4+x^3+x^2+x+1))
%F A194494 Explicit formula: n^8/384 - n^7/16 + 23*n^6/36 - 1301*n^5/360 + 14125*n^4/1152 - 8013*n^3/320 + 83147*n^2/2880 - 2089*n/160 + (n^4/32 - 17*n^3/24 + 193*n^2/32 - 2209*n/96 + 439/16)*floor(n/2) + (n^2/3 - 13*n/3 + 136/9)*floor(n/3) - 28/9*floor((n+1)/3) + 23/4*floor(n/4) - 3*floor((n+1)/4) + 4/5*floor(n/5) - 2/5*floor((n+2)/5) - 2/5*floor((n+3)/5), [_Vaclav Kotesovec_, Apr 08 2012]
%e A194494 Some solutions for 6X6
%e A194494 ..0............0............0............0............0............0
%e A194494 ..1.0..........0.0..........0.0..........0.1..........0.0..........0.1
%e A194494 ..0.0.0........1.0.0........0.0.1........0.0.0........0.1.0........0.0.0
%e A194494 ..0.1.0.0......0.0.1.0......1.0.0.0......1.0.0.0......0.0.0.1......0.0.1.0
%e A194494 ..0.0.0.0.1....0.0.0.0.1....0.0.0.1.0....0.0.1.0.0....1.0.0.0.0....1.0.0.0.0
%e A194494 ..0.0.1.0.0.0..0.1.0.0.0.0..0.1.0.0.0.0..0.0.0.0.1.0..0.0.1.0.0.0..0.0.0.1.0.0
%K A194494 nonn
%O A194494 1,6
%A A194494 _R. H. Hardin_ Aug 26 2011