A182407 -id:A182407 - cp's OEIS frontend

A141243 Number of ways to place non-attacking knights on the n X n board.

1, 2, 16, 94, 1365, 55213, 3368146, 394631712, 101693175442, 50929053498909, 48988729226134301, 96325314726538906164, 375615195988659173454092, 2933480442104347575000834468, 45480806737377995771543610802659, 1422902021111889804120495149240353936

Offset: 0

Max Alekseyev, Jun 17 2008

The maximum number of non-attacking knights is given by A030978.

Also the number of vertex covers and independent vertex sets in the n X n knight graph.

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69-72, 283-285.
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Knight Graph
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A182407, A182408, A201540.

Row sums of A244081.

b[n_, l_] := b[n, l] = Module[{d, f, g, k}, d = Length[l]/3; f = False; Which[n == 0, 1, l[[1 ;; d]] == Array[f &, d], b[n - 1, Join[l[[d + 1 ;; 3*d]], Array[True &, d]]], True, For[k = 1, ! l[[k]], k++]; g = ReplacePart[l, k -> f];
     If[k > 1, g = ReplacePart[g, 2*d - 1 + k -> f]];
     If[k < d, g = ReplacePart[g, 2*d + 1 + k -> f]];
     If[k > 2, g = ReplacePart[g, d - 2 + k -> f]];
     If[k < d - 1, g = ReplacePart[g, d + 2 + k -> f]];
     Expand[b[n, ReplacePart[l, k -> f]] + b[n, g]*x]]];
a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Array[True &, n*3]]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 10}] (* Jean-François Alcover, Mar 29 2016, after Alois P. Heinz's code for A244081 *)

A141243(n=4, s=n^2-1, bad=0)={ while(s && bittest(bad, s), s--);
   if(s < n, 2^(s+1-hammingweight(bad % (2<A141243(n, s-1, bad), x = s%n);
      x > 1 && bad = bitor(bad, 2^(s-n-2)); x < n-2 && bad = bitor(bad, 2^(s-n+2));
      if( s >= 2*n, x && bad = bitor(bad, 2^(s-2*n-1));
              x < n-1 && bad = bitor(bad, 2^(s-2*n+1))
   ); cnt + A141243(n, s-1, bad))} \\ M. F. Hasler, Mar 18 2025

def A141243(n=4, start=(1,1), forbidden=()):
    if start[0] >= n: return 2**sum((n,y+1) not in forbidden for y in range(n))
    nxt = (start[0],start[1]+1) if start[1]A141243(n, nxt, forbidden)
    if start in forbidden: return cnt
    forbidden = {s for s in forbidden if s >= nxt}
    if start[1]2: forbidden |= {(start[0]+1,start[1]-2)}
    if start[0]1: forbidden |= {(start[0]+2,start[1]-1)}
    return cnt+A141243(n, nxt, forbidden) # M. F. Hasler, Mar 17 2025

a(8)-a(13) from R. H. Hardin, Aug 25 2008
a(0) from Alois P. Heinz, Jun 19 2014
a(14) from Hiroaki Yamanouchi, Aug 28 2014
a(15) from Hiroaki Yamanouchi, Aug 29 2014

A212269 Number of ways to place k non-attacking kings on an n X n cylindrical chessboard, summed over all k >= 0.

Original entry on oeis.org

2, 5, 19, 205, 3011, 92875, 4763459, 459630701, 78223965193, 24270274906085, 13497818986883771, 13571363009654254429, 24562890586806439035377, 80199120146273882569630015, 471874707649862024071657639861, 5005895207027974222377733802848093

Offset: 1

Vaclav Kotesovec, May 12 2012

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 343.

Vaclav Kotesovec, Table of n, a(n) for n = 1..27
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.165

Cf. A063443 (n+1), A067958, A194650-A194654, A182407, A247413, A212269.

Limit n ->infinity (a(n))^(1/n^2) = 1.342643951124... (see A247413).

A212270 Number of ways to place k non-attacking wazirs on an n x n cylindrical chessboard, summed over all k >= 0.

Original entry on oeis.org

2, 7, 43, 933, 36211, 3557711, 746156517, 363549830913, 394677987525997, 974602314570939359, 5418730454986467701985, 68176187476467835406646029, 1936241516342334422813929891295, 124281423643836238320564876791634465, 18018270577720149773239661332878801006033

Offset: 1

Vaclav Kotesovec, May 12 2012

Wazir is a leaper [0,1].

Vaclav Kotesovec, Table of n, a(n) for n = 1..32
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.414

Main diagonal of A286513.

Cf. A006506, A027683, A182407, A212269, A212271.

Limit n ->infinity (a(n))^(1/n^2) is the hard square entropy constant A085850.

A212271 Number of ways to place k non-attacking ferses on an n x n cylindrical chessboard, summed over all k >= 0.

Original entry on oeis.org

2, 9, 80, 1600, 79033, 8156736, 2055960192, 1108756350625, 1411080429618656, 3943472747846953216, 25425527581172360096017, 365481944233773616212640000, 11980566143208960475692367828480, 882106482533191605447029340350009049, 147314997388032765439791110273770608260928

Offset: 1

Vaclav Kotesovec, May 12 2012

Fers is a leaper [1,1].

Vaclav Kotesovec, Table of n, a(n) for n = 1..18
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.443

Cf. A067965, A067960, A182407, A212269, A212270.

Limit n ->infinity (a(n))^(1/n^2) is the hard square entropy constant A085850.

A182408 Number of ways to place k non-attacking knights on an n x n toroidal chessboard, summed over all k >= 0.

Original entry on oeis.org

2, 7, 34, 743, 1546, 598078, 6027057, 10163241031, 242407820869

Offset: 1

Vaclav Kotesovec, May 09 2012

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.310

Cf. A141243, A182407, A067958, A027683, A067960.

cp's OEIS Frontend

A141243 Number of ways to place non-attacking knights on the n X n board.

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A212269 Number of ways to place k non-attacking kings on an n X n cylindrical chessboard, summed over all k >= 0.

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A212270 Number of ways to place k non-attacking wazirs on an n x n cylindrical chessboard, summed over all k >= 0.

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A212271 Number of ways to place k non-attacking ferses on an n x n cylindrical chessboard, summed over all k >= 0.

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A182408 Number of ways to place k non-attacking knights on an n x n toroidal chessboard, summed over all k >= 0.

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