A247413 -id:A247413 - 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

A085850 Decimal expansion of hard square entropy constant.

Original entry on oeis.org

1, 5, 0, 3, 0, 4, 8, 0, 8, 2, 4, 7, 5, 3, 3, 2, 2, 6, 4, 3, 2, 2, 0, 6, 6, 3, 2, 9, 4, 7, 5, 5, 5, 3, 6, 8, 9, 3, 8, 5, 7, 8, 1, 0

Offset: 1

Eric W. Weisstein, Jul 05 2003

			1.503048082...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.12, p. 342.

R. J. Baxter, Planar Lattice Gases with Nearest-Neighbour Exclusion, arXiv:cond-mat/9811264 [cond-mat.stat-mech], 1998; Annals of Combinatorics, 3 (1999), pp. 191-203.
S. R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat. vol 3 (1999) issue (2-4) pp. 323-335.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69 and 372.
Kai Liang, Independent Set Enumeration and Estimation of Related Constants of Grid Graphs, arXiv:2507.04007 [math.CO], 2025. See p. 8.
Eric Weisstein's World of Mathematics, Hard Square Entropy Constant.

Cf. A006506, A085851, A247413.

More terms from R. J. Mathar, Jul 18 2007

Terms a(19)-a(44) computed R. J. Baxter, 1998

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).

A085851 Decimal expansion of hard hexagon entropy constant.

Original entry on oeis.org

1, 3, 9, 5, 4, 8, 5, 9, 7, 2, 4, 7, 9, 3, 0, 2, 7, 3, 5, 2, 2, 9, 5, 0, 0, 6, 6, 3, 5, 6, 6, 8, 8, 8, 0, 6, 8, 9, 5, 4, 1, 0, 3, 7, 2, 8, 1, 4, 4, 6, 6, 1, 1, 9, 0, 8, 1, 7, 4, 7, 2, 1, 5, 6, 1, 3, 5, 7, 6, 0, 8, 8, 0, 3, 5, 8, 6, 9, 7, 7, 7, 4, 6, 8, 9, 8, 3, 7, 8, 7, 3, 0, 8, 5, 2, 7, 5, 4, 2, 7, 9, 0

Offset: 1

Eric W. Weisstein, Jul 05 2003

			1.395485972...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.12, p. 343.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69.
Eric Weisstein's World of Mathematics, Hard Hexagon Entropy Constant.

Cf. A085850, A247413, A066864.

RealDigits[N[Root[-32751691810479015985152 + 97143135277377575190528*#1^4 - 73347491183630103871488*#1^6 - 71220809441400405884928*#1^8 + 107155448150443388043264*#1^10 - 72405670285649161617408*#1^12 + 2958015038376958230528*#1^14 + 7449488310131083100160*#1^16 + 797726698866658379776*#1^18 + 2505062311720673792* #1^20 + 2013290651222784*#1^22 + 25937424601*#1^24 & , 2], 200]][[1]] (* Vaclav Kotesovec, Apr 03 2014 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A085850 Decimal expansion of hard square entropy constant.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Formula

A085851 Decimal expansion of hard hexagon entropy constant.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica