A212269 -id:A212269 - cp's OEIS frontend

A063443 Number of ways to tile an n X n square with 1 X 1 and 2 X 2 tiles.

1, 1, 2, 5, 35, 314, 6427, 202841, 12727570, 1355115601, 269718819131, 94707789944544, 60711713670028729, 69645620389200894313, 144633664064386054815370, 540156683236043677756331721, 3641548665525780178990584908643, 44222017282082621251230960522832336

Offset: 0

Reiner Martin, Jul 23 2001

a(n) is also the number of ways to populate an n-1 X n-1 chessboard with nonattacking kings (including the case of zero kings). Cf. A193580. - Andrew Woods, Aug 27 2011

Also the number of vertex covers and independent vertex sets of the n-1 X n-1 king graph.

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

Andrew Woods and Vaclav Kotesovec and Johan Nilsson, Table of n, a(n) for n = 0..40 (terms 0..21 from Andrew Woods, terms 22..24 from Vaclav Kotesovec and terms 25..40 from Johan Nilsson)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 68-69.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO], 2016, Section 4.1.
J. Nilsson, On Counting the Number of Tilings of a Rectangle with Squares of Size 1 and 2, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.2.
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A001045, A006506, A054854, A054855, A063650-A063653, A067966, etc.
Cf. A045846, A211348, A247413, A201513.
Cf. A212269, A067958.
a(n) = row sum n-1 of A193580.
Main diagonal of A245013.

Needs["LinearAlgebra`MatrixManipulation`"] Remove[mat] step[sa[rules1_, {dim1_, dim1_}], sa[rules2_, {dim2_, dim2_}]] := sa[Join[rules2, rules1 /. {x_Integer, y_Integer} -> {x + dim2, y}, rules1 /. {x_Integer, y_Integer} -> {x, y + dim2}], {dim1 + dim2, dim1 + dim2}] mat[0] = sa[{{1, 1} -> 1}, {1, 1}]; mat[1] = sa[{{1, 1} -> 1, {1, 2} -> 1, {2, 1} -> 1}, {2, 2}]; mat[n_] := mat[n] = step[mat[n - 2], mat[n - 1]]; A[n_] := mat[n] /. sa -> SparseArray; F[n_] := MatrixPower[A[n], n + 1][[1, 1]]; (* Mark McClure (mcmcclur(AT)bulldog.unca.edu), Mar 19 2006 *)
$RecursionLimit = 1000; Clear[a, b]; b[n_, l_List] := b[n, l] = Module[{m=Min[l], k}, If[m>0, b[n-m, l-m], If[n == 0, 1, k=Position[l, 0, 1, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k 2, k+1 -> 2}]], 0]]]]; a[n_] := a[n] = If[n<2, 1, b[n, Table[0, {n}]]]; Table[Print[a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 11 2014, after Alois P. Heinz *)

Lim_{n -> infinity} (a(n))^(1/n^2) = A247413 = 1.342643951124... . - Brendan McKay, 1996

4 more terms from R. H. Hardin, Jan 23 2002
2 more terms from Keith Schneider (kschneid(AT)bulldog.unca.edu), Mar 19 2006
5 more terms from Andrew Woods, Aug 27 2011
a(22)-a(24) in b-file from Vaclav Kotesovec, May 01 2012
a(0) inserted by Alois P. Heinz, Sep 17 2014
a(25)-a(40) in b-file from Johan Nilsson, Mar 10 2016

A067958 Number of binary arrangements without adjacent 1's on n X n torus connected e-w ne-sw n-s nw-se.

Original entry on oeis.org

1, 5, 10, 133, 1411, 42938, 1796859, 157763829, 22909432780, 6291183426165, 3032485231813445, 2674030233698391466, 4216437656471537450175, 12038380931111061789962901, 61810608197507432888286102310, 572863067272579464080483552434421

Offset: 1

R. H. Hardin, Feb 02 2002

For n > 1, a(n) is also the number of ways to populate an n X n toroidal chessboard with non-attacking kings (including the case of zero kings). - Vaclav Kotesovec, Oct 10 2011

			Neighbors for n=4:
  :\|/\|/\|/\|/
  :-o--o--o--o-
  :/|\/|\/|\/|\
  :\|/\|/\|/\|/
  :-o--o--o--o-
  :/|\/|\/|\/|\
  :\|/\|/\|/\|/
  :-o--o--o--o-
  :/|\/|\/|\/|\
  :\|/\|/\|/\|/
  :-o--o--o--o-
  :/|\/|\/|\/|\

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 214.

Cf. circle A000204, line A000045, arrays: ne-sw nw-se A067965, e-w ne-sw nw-se A067963, n-s nw-se A067964, e-w n-s nw-se A066864, e-w ne-sw n-s nw-se A063443, n-s A067966, e-w n-s A006506, nw-se A067962, toruses: bare A002416, ne-sw nw-se A067960, ne-sw n-s nw-se A067959, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.

Cf. A212269.

a(14) from Vaclav Kotesovec, Aug 22 2016

a(15)-a(16) from Vaclav Kotesovec, May 15 2021

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.

A247413 Decimal expansion of the entropy constant related to A063443.

Original entry on oeis.org

1, 3, 4, 2, 6, 4, 3, 9, 5, 1, 1, 2, 4

Offset: 1

Vaclav Kotesovec, Sep 16 2014

			1.342643951124...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 343.
B. D. McKay, On Calkin and Wilf's limit theorem for grid graphs, unpublished note, 1996.

N. J. Calkin, K. James, and S. Purvis, Counting Kings, 2006, p.9.
N. J. Calkin and H. S. Wilf, The number of independent sets in a grid graph, preprint, 1995.
N. J. Calkin and H. S. Wilf, The number of independent sets in a grid graph, SIAM J. Discrete Math, 11 (1998) 54-60.
Steven R. Finch, Several Constants Arising in Statistical Mechanics, arXiv:math/9810155 [math.CO], 1999, p.8.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.68-69.
D. E. Knuth, Nonattacking kings on a chessboard, 1994.
M. Larsen, The Problem of Kings, The Electronic Journal of Combinatorics 2, 1995.
Kai Liang, Independent Set Enumeration in King Graphs by Tensor Network Contractions, arXiv:2505.12776 [math.CO], 2025. See p. 6.
Eric Weisstein's World of Mathematics, Hard Hexagon Entropy Constant
Eric Weisstein's World of Mathematics, Hard Square Entropy Constant
H. S. Wilf, The problem of the kings, Elec. J. Combin. 2, 1995.

Cf. A063443, A212269.

Cf. A085850, A085851.

Equals limit n->infinity (A063443(n))^(1/n^2).

Equals limit n->infinity (A212269(n))^(1/n^2).

cp's OEIS Frontend

A063443 Number of ways to tile an n X n square with 1 X 1 and 2 X 2 tiles.

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A067958 Number of binary arrangements without adjacent 1's on n X n torus connected e-w ne-sw n-s nw-se.

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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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A247413 Decimal expansion of the entropy constant related to A063443.

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