A006506 -id:A006506 - cp's OEIS frontend

A085850 Decimal expansion of hard square entropy constant.

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

A181031 Array read by antidiagonals: T(n,k) = number of n X k binary matrices with no initial bit string in any row or column divisible by 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 5, 1, 1, 8, 17, 17, 8, 1, 1, 13, 41, 63, 41, 13, 1, 1, 21, 99, 227, 227, 99, 21, 1, 1, 34, 239, 827, 1234, 827, 239, 34, 1, 1, 55, 577, 2999, 6743, 6743, 2999, 577, 55, 1, 1, 89, 1393, 10897, 36787, 55447, 36787, 10897, 1393, 89, 1, 1, 144

Offset: 1

R. H. Hardin, Sep 30 2010

A089980 is essentially the same array (see the Comments by Steve Butler in A006506). - N. J. A. Sloane, Jan 27 2015

			Table starts
.1..1....1......1.......1.........1...........1............1..............1
.1..2....3......5.......8........13..........21...........34.............55
.1..3....7.....17......41........99.........239..........577...........1393
.1..5...17.....63.....227.......827........2999........10897..........39561
.1..8...41....227....1234......6743.......36787.......200798........1095851
.1.13...99....827....6743.....55447......454385......3729091.......30584687
.1.21..239...2999...36787....454385.....5598861.....69050253......851302029
.1.34..577..10897..200798...3729091....69050253...1280128950....23720149995
.1.55.1393..39561.1095851..30584687...851302029..23720149995...660647962955
.1.89.3363.143677.5980913.250916131.10496827403.439621976195.18403310404291
Some solutions for 5X5
..1..1..1..1..1....1..1..1..1..1....1..1..1..1..1....1..1..1..1..1
..1..0..1..0..1....1..0..1..0..1....1..0..1..0..1....1..0..1..0..1
..1..1..0..1..0....1..1..0..1..0....1..1..0..1..0....1..1..0..1..0
..1..0..1..0..1....1..0..1..0..1....1..0..1..0..1....1..0..1..0..1
..1..1..0..1..0....1..1..0..1..1....1..1..1..1..0....1..1..1..1..1

R. H. Hardin, Table of n, a(n) for n=1..1057

Cf. A089980.

A066865 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal torus bent for odd n.

Original entry on oeis.org

1, 5, 22, 217, 4726, 164258, 14840533, 1834600977, 669877863205, 296979228487760, 434542100979981567, 692625866382651263578, 4053364289624915167879497, 23237986479606982160703729647, 543749373021017146939376423644362, 11213018647250714014261414954480048385

Offset: 1

R. H. Hardin, Jan 25 2002

nonn

			Neighbors for n=4:
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
Neighbors for n=5:
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.
J. Katzenelson and R. P. Kurshan, S/R: A Language for Specifying Protocols and Other Coordinating Processes, pp. 286-292 in Proc. IEEE Conf. Comput. Comm., 1986.

Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]

Cf. A006506, A027683, A066863, A066864, A066866, A067967 (shifted instead of bent).

Row sums of A067015.

More terms from Sean A. Irvine, Nov 18 2023

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.

A066863 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal grid.

Original entry on oeis.org

2, 6, 43, 557, 14432, 719469, 70372090, 13351521479, 4941545691252, 3559349503024593, 4993739972681894885, 13642580224488264353504, 72582736229683196932680697, 751993955499337790653321567382, 15172223086707160824288341875907978

Offset: 1

R. H. Hardin, Jan 25 2002

nonn

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

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.
J. Katzenelson and R. P. Kurshan, S/R: A Language for Specifying Protocols and Other Coordinating Processes, pp. 286-292 in Proc. IEEE Conf. Comput. Comm., 1986.

Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]
Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Hard Hexagon Entropy Constant

Cf. A006506, A027683, A066864, A066865, A066866.

More terms from Sean A. Irvine, Nov 15 2023

A089980 Array read by antidiagonals: T(n,m) = number of independent sets in the grid graph P_n X P_m.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 5, 1, 1, 8, 17, 17, 8, 1, 1, 13, 41, 63, 41, 13, 1, 1, 21, 99, 227, 227, 99, 21, 1, 1, 34, 239, 827, 1234, 827, 239, 34, 1, 1, 55, 577, 2999, 6743, 6743, 2999, 577, 55, 1, 1, 89, 1393, 10897, 36787, 55447, 36787, 10897, 1393, 89, 1

Offset: 0

Mitch Harris, Nov 17 2003

This table is indexed starting at 0. The table in A089934 is 1 based.

A181031 is essentially the same array (see the Comments by Steve Butler in A006506). - N. J. A. Sloane, Jan 27 2015

			Square array T(n,m) begins:
  1,  1,  1,   1,    1,     1, ...
  1,  2,  3,   5,    8,    13, ...
  1,  3,  7,  17,   41,    99, ...
  1,  5, 17,  63,  227,   827, ...
  1,  8, 41, 227, 1234,  6743, ...
  1, 13, 99, 827, 6743, 55447, ...

Liang Kai, Antidiagonals n = 0..78, flattened
Kai Liang, Independent Set Enumeration in King Graphs by Tensor Network Contractions, arXiv:2505.12776 [math.CO], 2025. See p. 1.
Kai Liang, Independent Set Enumeration and Estimation of Related Constants of Grid Graphs, arXiv:2507.04007 [math.CO], 2025. See p. 1.

Main entry: A089934.
Main diagonal gives A006506.
Cf. A181031.

A054867 Number of non-attacking configurations on a diamond of size n, where a prince attacks the four adjacent non-diagonal squares.

Original entry on oeis.org

1, 2, 17, 689, 139344, 142999897, 748437606081, 19999400591072512, 2728539172202554958697, 1900346273206544901717879089, 6755797872872106084596492075448192, 122584407857548123729431742141838309441329, 11352604691637658946858196503018301306800588837281

Offset: 0

Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000

A diamond of size n X n contains (n^2 + (n-1)^2) = A001844(n-1) squares.

For n > 0, a(n) is the number of ways to place non-adjacent counters on the black squares of a 2n-1 X 2n-1 checker board. The checker board is such that the black squares are in the corners. - Andrew Howroyd, Jan 16 2020

			From _Andrew Howroyd_, Jan 16 2020: (Start)
Case n=2: The grid consists of 5 squares as shown below.
        __
     __|__|__
    |__|__|__|
       |__|
If a prince is placed on the central square then a prince cannot be placed on the other 4 squares, otherwise princes can be placed in any combination. The total number of non-attacking configurations is then 1 + 2^4 = 17, so a(2) = 17.
.
Case n=3: The grid consists of 13 squares as shown below:
           __
        __|__|__
     __|__|__|__|__
    |__|__|__|__|__|
       |__|__|__|
          |__|
The total number of non-attacking configurations of princes is 689 so a(3) = 689.
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..20
Eric Weisstein's World of Mathematics, Independent Vertex Set

Main diagonal of A331406.

Cf. A006506, A001844.

a(0)=1 prepended and terms a(5) and beyond from Andrew Howroyd, Jan 15 2020

A197048 Number of n X n 0..4 arrays with each element equal to the number of its horizontal and vertical zero neighbors.

Original entry on oeis.org

1, 2, 10, 42, 358, 4468, 88056, 2745186, 134355866, 10264692132, 1234801357470, 232966546265096, 68939282741912248

Offset: 1

R. H. Hardin, Oct 09 2011

nonn

Every 0 is next to 0 0's, every 1 is next to 1 0's, every 2 is next to 2 0's, every 3 is next to 3 0's, every 4 is next to 4 0's.

Also, the number of maximal independent vertex sets in the grid graph P_n X P_n. - Andrew Howroyd, May 16 2017

			Some solutions for n=4
..0..2..0..2....2..0..1..1....2..0..3..0....0..3..0..2....1..0..3..0
..1..1..2..0....0..3..1..0....0..4..0..2....3..0..3..0....1..2..0..3
..2..0..2..1....3..0..2..1....3..0..2..1....0..2..1..1....0..1..3..0
..0..3..0..1....0..3..0..1....0..2..1..0....1..1..0..1....1..1..0..2

Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Diagonal of A197054.

Cf. A006506 (independent vertex sets), A133515 (dominating sets).

A197054 = Cases[Import["https://oeis.org/A197054/b197054.txt", "Table"], {, }][[All, 2]];
a[n_] := A197054[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Sep 23 2019 *)

A050974 Number of binary arrangements on n X n array without three adjacent 1's in a row or column.

Original entry on oeis.org

1, 2, 16, 265, 16561, 3157010, 1828904402, 3323590649777, 18691199385898465, 325778072452564800064, 17617718915229579206450786, 2954164381835835259001326344913, 1536134628973698280539373190731911729, 2477137610106747308204461168746042225266836, 12387488188151269567355592399321080831513078632498, 192102098800681202990688566451981906679020804069237862571, 9238409697848267958752630399467598421213391733838644131510525089

Offset: 0

Eric W. Weisstein

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.

Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]
Eric Weisstein's World of Mathematics, 01-Matrix.

Any connected three 1's gives A067968.

Cf. A006506. Diagonal of A202471.

t[m_] := t[m] = Map[ArrayReshape[#, {m, m}] &, Tuples[{0, 1}, m^2]]; a[m_] := a[m] = Count[Table[AnyTrue[Flatten[{Table[Equal[1, t[m][[n, a, b]], t[m][[n, a, b + 1]], t[m][[n, a, b + 2]]], {a, 1, m}, {b, 1, m - 2}], Table[Equal[1, t[m][[n, a, b]], t[m][[n, a + 1, b]], t[m][[n, a + 2, b]]], {a, 1, m - 2}, {b, 1, m}]}], TrueQ], {n, 1, 2^(m^2)}], False]; (* Robert P. P. McKone, Jan 04 2022 *)

More terms from R. H. Hardin, Feb 02 2002

a(0)=1 prepended and a(13)-a(16) from Peter J. Taylor, Sep 26 2024

A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

Original entry on oeis.org

2, 16, 288, 11664, 1458000, 506250000, 414720000000, 869730877440000, 5045702916833280000, 77297454895962562560000, 3017525202366485003182080000, 307389127582207654481154908160000, 83016370640108703579427655610531840000, 58770343311359208383258439665073059266560000

Offset: 1

Vaclav Kotesovec, May 05 2012

nonn

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

Vaclav Kotesovec, Table of n, a(n) for n = 1..60
V. Kotesovec, Non-attacking chess pieces

Cf. A067962, A067966, A063443, A006506, A067965, A066864, A067963, A067964, A182563

Table[If[EvenQ[n],Fibonacci[n/2+2]^(n+2)*Product[Fibonacci[j+2]^4,{j,1,n/2-1}],Fibonacci[(n+1)/2+2]^((n+1)/2)*Fibonacci[(n-1)/2+2]^((n-1)/2)*Product[Fibonacci[j+2]^4,{j,1,(n-1)/2}]],{n,1,20}]

a(n) = F(n/2+2)^(n+2)*prod(j=1,n/2-1,F(j+2)^4) if n is even, F((n+1)/2+2)^((n+1)/2)*F((n-1)/2+2)^((n-1)/2)*prod(j=1,(n-1)/2,F(j+2)^4) if n is odd, where F(n) = A000045(n) is the n-th Fibonacci number.

a(n) is asymptotic to C^4*((1+sqrt(5))/2)^((n+2)*(n+4))/5^(3/2*(n+2)), where C=1.226742010720353244... is Fibonacci Factorial Constant, see A062073.

cp's OEIS Frontend

A085850 Decimal expansion of hard square entropy constant.

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A181031 Array read by antidiagonals: T(n,k) = number of n X k binary matrices with no initial bit string in any row or column divisible by 4.

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A066865 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal torus bent for odd n.

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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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A066863 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal grid.

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A089980 Array read by antidiagonals: T(n,m) = number of independent sets in the grid graph P_n X P_m.

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A054867 Number of non-attacking configurations on a diamond of size n, where a prince attacks the four adjacent non-diagonal squares.

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Comments

Examples

Links

Crossrefs

Extensions

A197048 Number of n X n 0..4 arrays with each element equal to the number of its horizontal and vertical zero neighbors.

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Programs

Mathematica

A050974 Number of binary arrangements on n X n array without three adjacent 1's in a row or column.

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Mathematica