A193986 -id:A193986 - cp's OEIS frontend

A283113 Triangle read by rows: T(n,k) is the number of nonequivalent ways (mod D_3) to place k points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal (n >= 1).

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 5, 1, 5, 19, 23, 3, 1, 7, 38, 82, 40, 1, 1, 8, 66, 230, 242, 45, 1, 10, 110, 560, 1038, 533, 29, 1, 12, 170, 1208, 3504, 3546, 821, 6, 1, 14, 255, 2392, 9998, 16917, 9137, 807, 1, 16, 365, 4405, 25158, 64345, 63755, 17408, 422

Offset: 1

Heinrich Ludwig, Mar 10 2017

Length of n-th row is A004396(n) + 1, for 1 <= n <= 21, where A004396(n) is the maximal number of points that can be placed under the condition mentioned above.
Rotations and reflections of placements are not counted. If they are to be counted, see A193986.
In terms or triangular chess: Number of nonequivalent ways (mod D_3) to arrange k nonattacking rooks on an n X n X n board, k>=0, n>=1.

			The table begins with T(1,0), T(1,1);
  1,  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,   1;
  1,  4,   9,   5;
  1,  5,  19,  23,    3;
  1,  7,  38,  82,   40,   1;
  1,  8,  66, 230,  242,  45;
  1, 10, 110, 560, 1038, 533, 29;
  ...

Heinrich Ludwig, Table of n, a(n) for n = 1..175

Row sums give A283117.
Columns 2..6: A001399, A005994, A283114, A283115, A283116.
Cf. A004396, A193986.

A289709 Number of independent vertex sets and vertex covers in the n-triangular honeycomb queen graph.

Original entry on oeis.org

2, 4, 10, 28, 84, 272, 946, 3486, 13560, 55432, 236852, 1054928, 4881972, 23420436, 116204016, 595246848, 3142169416, 17068245184, 95267426432, 545732236936, 3204607199704

Offset: 1

Eric W. Weisstein, Jul 14 2017

Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A193986, A283117, A289883, A289893, A326611.

From Andrew Howroyd, Sep 12 2019: (Start)
a(n) = 6*A283117(n) - 2*A326611(n) - 3*2^ceiling(n/2).
a(n) = 1 + Sum_{k>=1} A193986(n,k).
(End)

a(13)-a(21) from Andrew Howroyd, Sep 12 2019

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

Original entry on oeis.org

0, 0, 0, 2, 23, 127, 468, 1352, 3310, 7190, 14260, 26330, 45885, 76237, 121688, 187712, 281148, 410412, 585720, 819330, 1125795, 1522235, 2028620, 2668072, 3467178, 4456322, 5670028, 7147322, 8932105, 11073545, 13626480, 16651840, 20217080

Offset: 1

R. H. Hardin Aug 10 2011

nonn

Column 3 of A193986

			Some solutions for 5X5X5
......0..........0..........0..........0..........0..........0..........0
.....0.0........0.0........0.0........0.0........0.1........0.0........0.1
....0.0.1......1.0.0......0.1.0......0.1.0......0.0.0......0.1.0......1.0.0
...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
..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

R. H. Hardin, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.

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)
Contribution from Vaclav Kotesovec, Aug 31 2012: (Start)
Empirical: G.f.: -x^4*(2 + 11*x + 17*x^2)/((-1+x)^7*(1+x))
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)
(End)

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

Original entry on oeis.org

0, 0, 0, 0, 0, 18, 233, 1449, 6213, 20993, 59943, 150903, 344323, 726033, 1434678, 2685046, 4798206, 8240022, 13669026, 21995586, 34453386, 52685556, 78846471, 115721991, 166869131, 236778399, 331059729, 456655745, 622083189, 837706779

Offset: 1

R. H. Hardin Aug 10 2011

nonn

Column 4 of A193986

			Some solutions for 6X6X6
.......0............0............0............0............0............0
......0.0..........0.0..........1.0..........0.0..........0.1..........0.0
.....0.0.1........1.0.0........0.0.0........0.1.0........1.0.0........1.0.0
....0.1.0.0......0.0.0.1......0.0.0.1......0.0.0.1......0.0.0.0......0.0.1.0
...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
..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

R. H. Hardin, Table of n, a(n) for n = 1..89
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.

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)
Contribution from Vaclav Kotesovec, Aug 31 2012: (Start)
Empirical: G.f.: -x^6*(18 + 125*x + 267*x^2 + 279*x^3 + 151*x^4)/((-1+x)^9*(1+x)^3)
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)
(End)

A193983 Number of ways to arrange 5 nonattacking triangular rooks on an n X n X n triangular grid.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 270, 3195, 21273, 101484, 386052, 1243899, 3527469, 9035376, 21297492, 46838142, 97131762, 191517192, 361427508, 656353494, 1152094086, 1961910990, 3251400894, 5257953789, 8315944731, 12888836064, 19609755396

Offset: 1

R. H. Hardin Aug 10 2011

nonn

			Some solutions for 7 X 7 X 7
........0..............0..............0..............0..............0
.......0.0............0.0............0.0............0.0............0.0
......0.1.0..........0.1.0..........0.0.1..........1.0.0..........0.0.1
.....1.0.0.0........0.0.0.1........1.0.0.0........0.0.0.1........0.1.0.0
....0.0.0.0.1......1.0.0.0.0......0.0.0.1.0......0.1.0.0.0......1.0.0.0.0
...0.0.0.1.0.0....0.0.1.0.0.0....0.1.0.0.0.0....0.0.0.0.1.0....0.0.0.0.1.0
..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

R. H. Hardin, Table of n, a(n) for n = 1..52
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.

Column 5 of A193986.

Empirical: a(n) = 5*a(n-1) -5*a(n-2) -14*a(n-3) +30*a(n-4) +6*a(n-5) -50*a(n-6) +10*a(n-7) +44*a(n-8) -44*a(n-10) -10*a(n-11) +50*a(n-12) -6*a(n-13) -30*a(n-14) +14*a(n-15) +5*a(n-16) -5*a(n-17) +a(n-18).
Contribution from Vaclav Kotesovec, Aug 31 2012: (Start)
Empirical: G.f.: -3*x^7*(2 + 80*x + 625*x^2 + 2244*x^3 + 4898*x^4 + 7197*x^5 + 7237*x^6 + 5030*x^7 + 2294*x^8 + 633*x^9)/((-1+x)^11*(1+x)^5*(1+x+x^2)).
Empirical: a(n) = 3461*n/320 - 469*n^2/240 - 469*n^3/15 + 2383*n^4/64 - 76607*n^5/3840 + 23693*n^6/3840 - 2263*n^7/1920 + 53*n^8/384 - 7*n^9/768 + n^10/3840 + 4/3*floor(n/3) + (1359/32 - 247*n/8 + 245*n^2/32 - 13*n^3/16 + n^4/32)*floor(n/2) - 4/3*floor((1 + n)/3).
(End)

A193984 Number of ways to arrange 6 nonattacking triangular rooks on an n X n X n triangular grid.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 166, 4902, 54771, 382439, 1976455, 8250687, 29309540, 91705972, 258870740, 671005444, 1618468198, 3670491998, 7891420850, 16191666766, 31878943876, 60501909500, 111108316262, 198083991586, 343784805209

Offset: 1

R. H. Hardin Aug 10 2011

nonn

			Some solutions for 9 X 9 X 9
..........0..................0..................0..................0
.........0.0................0.0................0.0................0.1
........1.0.0..............0.0.0..............0.0.1..............0.0.0
.......0.0.0.0............1.0.0.0............0.0.0.0............1.0.0.0
......0.0.0.1.0..........0.0.0.1.0..........1.0.0.0.0..........0.0.0.0.0
.....0.0.1.0.0.0........0.1.0.0.0.0........0.0.0.1.0.0........0.0.0.1.0.0
....0.0.0.0.0.0.1......0.0.0.0.0.0.1......0.1.0.0.0.0.0......0.0.0.0.0.1.0
...0.1.0.0.0.0.0.0....0.0.0.0.0.1.0.0....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..0.0.1.0.0.0.0.0.0..0.0.0.0.0.0.0.1.0..0.0.0.0.1.0.0.0.0

R. H. Hardin, Table of n, a(n) for n = 1..37
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.

Column 6 of A193986.

Contribution from Vaclav Kotesovec, Aug 31 2012: (Start)
Empirical: Recurrence: a(n-29) - 4*a(n-28) + 17*a(n-26) - 9*a(n-25) - 32*a(n-24) + 7*a(n-23) + 51*a(n-22) + 26*a(n-21) - 77*a(n-20) - 59*a(n-19) + 58*a(n-18) + 74*a(n-17) + 21*a(n-16) - 74*a(n-15) - 74*a(n-14) + 21*a(n-13) + 74*a(n-12) + 58*a(n-11) - 59*a(n-10) - 77*a(n-9) + 26*a(n-8) + 51*a(n-7) + 7*a(n-6) - 32*a(n-5) - 9*a(n-4) + 17*a(n-3) - 4*a(n-1) + a(n) = 0.
Empirical: G.f.: -x^9*(166 + 4404*x + 39567*x^2 + 205744*x^3 + 734283*x^4 + 1960827*x^5 + 4120441*x^6 + 7036145*x^7 + 9956248*x^8 + 11823233*x^9 + 11839707*x^10 + 10002936*x^11 + 7077533*x^12 + 4145811*x^13 + 1957821*x^14 + 721991*x^15 + 191674*x^16 + 31709*x^17)/((-1+x)^13*(1+x)^7*(1+x^2)*(1+x+x^2)^3).
Empirical: a(n) = 98227*n/10080 + 39907*n^2/180 - 1105267*n^3/1920 + 6516731*n^4/11520 - 7025857*n^5/23040 + 4788163*n^6/46080 - 3842803*n^7/161280 + 34619*n^8/9216 - 9299*n^9/23040 + 29*n^10/1024 - 3*n^11/2560 + n^12/46080 + 25/4*floor(n/4) + (56 - 38*n/3 + 2*n^2/3)*floor(n/3) + (12053/16 - 32515*n/48 + 22687*n^2/96 - 8249*n^3/192 + 279*n^4/64 - 15*n^5/64 + n^6/192)*floor(n/2) - 3*floor((1+n)/4) + (-184/3 + 38*n/3 - 2*n^2/3)*floor((1+n)/3).
(End)

A193985 Number of ways to arrange 7 nonattacking triangular rooks on an nXnXn triangular grid.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 4842, 104448, 1127473, 8147469, 44813100, 201616740, 776296572, 2638333236, 8091512680, 22767562704, 59525327634, 146063694310, 339101167392, 749761416480, 1587403977944, 3233026755180, 6358705091310

Offset: 1

R. H. Hardin Aug 10 2011

nonn

Column 7 of A193986

			Some solutions for 10X10X10
...........0....................0....................0
..........0.0..................0.0..................0.0
.........0.0.0................0.0.0................0.0.0
........1.0.0.0..............0.1.0.0..............0.0.0.1
.......0.0.0.0.1............0.0.0.0.1............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........0.0.1.0.0.0.0........1.0.0.0.0.0.0
....0.0.0.0.0.0.1.0......0.0.0.0.0.0.1.0......0.0.0.0.0.0.1.0
...0.0.1.0.0.0.0.0.0....0.0.0.0.0.1.0.0.0....0.0.0.0.0.1.0.0.0
..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

R. H. Hardin, Table of n, a(n) for n = 1..30

cp's OEIS Frontend

A283113 Triangle read by rows: T(n,k) is the number of nonequivalent ways (mod D_3) to place k points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal (n >= 1).

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A289709 Number of independent vertex sets and vertex covers in the n-triangular honeycomb queen graph.

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

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

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A193983 Number of ways to arrange 5 nonattacking triangular rooks on an n X n X n triangular grid.

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A193984 Number of ways to arrange 6 nonattacking triangular rooks on an n X n X n triangular grid.

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A193985 Number of ways to arrange 7 nonattacking triangular rooks on an nXnXn triangular grid.

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