A194498 -id:A194498 - cp's OEIS frontend

A194493 Number of ways to arrange 3 nonattacking queens on the lower triangle of an n X n board.

0, 0, 0, 0, 12, 82, 330, 1008, 2566, 5742, 11652, 21926, 38802, 65322, 105428, 164214, 248022, 364764, 523998, 737334, 1018488, 1383768, 1852104, 2445628, 3189660, 4113396, 5249848, 6636636, 8315880, 10335110, 12747090, 15610860, 18991490

Offset: 1

R. H. Hardin Aug 26 2011

nonn

Column 3 of A194498

			Some solutions for 5X5
..0..........0..........0..........0..........0..........0..........1
..1.0........0.1........1.0........0.0........1.0........0.1........0.0
..0.0.0......0.0.0......0.0.0......1.0.0......0.0.1......0.0.0......0.1.0
..0.1.0.0....1.0.0.0....0.0.0.1....0.0.0.1....0.0.0.0....1.0.0.0....0.0.0.0
..0.0.0.0.1..0.0.1.0.0..0.1.0.0.0..0.1.0.0.0..0.1.0.0.0..0.0.0.1.0..0.0.1.0.0

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = 3*a(n-1) -7*a(n-3) +3*a(n-4) +6*a(n-5) -6*a(n-7) -3*a(n-8) +7*a(n-9) -3*a(n-11) +a(n-12), [R. H. Hardin Aug 26 2011]
G.f.: -2*x^5*(18*x^5 + 40*x^4 + 51*x^3 + 42*x^2 + 23*x + 6)/((x-1)^7*(x+1)^3*(x^2+x+1))
Explicit formula: n^6/48 - 11*n^5/48 + 15*n^4/16 - 241*n^3/144 + 17*n^2/16 - 17*n/144 + (n^2/8 - 9*n/8 + 17/8)*floor(n/2) + 2/3*floor(n/3), [Vaclav Kotesovec, Apr 08 2012]

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

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 118, 802, 3708, 13280, 39734, 104000, 244948, 530632, 1072776, 2048056, 3723314, 6492342, 10915254, 17777372, 28147380, 43465356, 65624634, 97098802, 141050688, 201509798, 283514112, 393348562, 538725268, 729098516

Offset: 1

R. H. Hardin Aug 26 2011

nonn

Column 4 of A194498

			Some solutions for 6X6
..0............0............0............0............0............0
..1.0..........0.0..........0.0..........0.1..........0.0..........0.1
..0.0.0........1.0.0........0.0.1........0.0.0........0.1.0........0.0.0
..0.1.0.0......0.0.1.0......1.0.0.0......1.0.0.0......0.0.0.1......0.0.1.0
..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
..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

R. H. Hardin, Table of n, a(n) for n = 1..81

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]
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))
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]

A194495 Number of ways to arrange 5 nonattacking queens on the lower triangle of an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 114, 1384, 9890, 50662, 205512, 698688, 2074530, 5525902, 13476246, 30522678, 64968996, 131070600, 252492296, 467017054, 833501728, 1440839952, 2420878990, 3964168632, 6342428908, 9934738024, 15264432954

Offset: 1

R. H. Hardin Aug 26 2011

nonn

Column 5 of A194498

			All solutions for 7X7
..0..............0..............0..............0
..0.0............0.0............0.0............0.0
..1.0.0..........0.1.0..........1.0.0..........0.0.1
..0.0.0.1........0.0.0.1........0.0.1.0........1.0.0.0
..0.1.0.0.0......1.0.0.0.0......0.0.0.0.1......0.0.0.1.0
..0.0.0.0.1.0....0.0.1.0.0.0....0.1.0.0.0.0....0.1.0.0.0.0
..0.0.1.0.0.0.0..0.0.0.0.1.0.0..0.0.0.1.0.0.0..0.0.0.0.1.0.0

R. H. Hardin, Table of n, a(n) for n = 1..49

A194496 Number of ways to arrange 6 nonattacking queens on the lower triangle of an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 64, 1644, 19306, 146718, 820218, 3670288, 13846830, 45661556, 134896662, 364019248, 909633276, 2129318414, 4709083054, 9912162234, 19970298696, 38707229282, 72458290216, 131476840106, 231906957348

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 6 of A194498.

			Some solutions for 9 X 9:
  0                  0                  0                  0
  0 1                0 0                1 0                0 0
  0 0 0              1 0 0              0 0 0              0 1 0
  1 0 0 0            0 0 0 0            0 0 0 0            0 0 0 0
  0 0 0 0 0          0 0 0 0 1          0 0 0 0 1          1 0 0 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
  0 0 0 0 0 1 0      0 0 0 0 0 1 0      0 0 0 1 0 0 0      0 0 0 0 1 0 0
  0 0 1 0 0 0 0 0    0 0 0 1 0 0 0 0    0 0 0 0 0 1 0 0    0 0 0 0 0 0 0 1
  0 0 0 0 1 0 0 0 0  0 1 0 0 0 0 0 0 0  0 0 1 0 0 0 0 0 0  0 0 0 1 0 0 0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..35

A194497 Number of ways to arrange 7 nonattacking queens on the lower triangle of an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1210, 27198, 322782, 2564988, 15372702, 74615814, 306900410, 1105651074, 3569169990, 10512880400, 28628691842, 72880455042, 174895586328, 398556484840, 867314768878, 1811576604968, 3646385889888

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 7 of A194498.

			Some solutions for 10 X 10:
  0                    0                    0
  0 0                  0 0                  0 0
  0 0 0                0 0 0                0 0 0
  0 1 0 0              0 1 0 0              0 0 0 1
  0 0 0 1 0            0 0 0 1 0            1 0 0 0 0
  1 0 0 0 0 0          0 0 0 0 0 1          0 0 1 0 0 0
  0 0 0 0 0 0 1        1 0 0 0 0 0 0        0 0 0 0 0 1 0
  0 0 0 0 1 0 0 0      0 0 1 0 0 0 0 0      0 1 0 0 0 0 0 0
  0 0 1 0 0 0 0 0 0    0 0 0 0 1 0 0 0 0    0 0 0 0 0 0 1 0 0
  0 0 0 0 0 1 0 0 0 0  0 0 0 0 0 0 1 0 0 0  0 0 0 0 1 0 0 0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..28

cp's OEIS Frontend

A194493 Number of ways to arrange 3 nonattacking queens on the lower triangle of an n X n board.

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

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A194495 Number of ways to arrange 5 nonattacking queens on the lower triangle of an n X n board.

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A194496 Number of ways to arrange 6 nonattacking queens on the lower triangle of an n X n board.

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A194497 Number of ways to arrange 7 nonattacking queens on the lower triangle of an n X n board.

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