A232833 -id:A232833 - cp's OEIS frontend

A006506 Number of n X n binary matrices with no 2 adjacent 1's, or number of configurations of non-attacking princes on an n X n board, where a "prince" attacks the four adjacent (non-diagonal) squares. Also number of independent vertex sets in an n X n grid.

1, 2, 7, 63, 1234, 55447, 5598861, 1280128950, 660647962955, 770548397261707, 2030049051145980050, 12083401651433651945979, 162481813349792588536582997, 4935961285224791538367780371090, 338752110195939290445247645371206783, 52521741712869136440040654451875316861275

Offset: 0

N. J. A. Sloane, R. H. Hardin, Paul Zimmermann

A two-dimensional generalization of the Fibonacci numbers.
Also the number of vertex covers in the n X n grid graph P_n X P_n.
A181030 (Number of n X n binary matrices with no leading bitstring in any row or column divisible by 4) is the same sequence. Proof from Steve Butler, Jan 26 2015: This is trivially true. A181030 is equivalent to this sequence by interchanging the roles of 0 and 1. In particular, A181030 looks for binary matrices with no leading bitstring divisible by 4, but a bitstring is divisible by 4 if and only if its last two digits is 0; in a binary matrix this can only be avoided if there are no two adjacent 0's (i.e., for any two adjacent 0's take the bitstring starting in that row or column and we are done); the present sequence looks for no two adjacent 1's. Similar reasons show that the array A181031 is equivalent to the array A089980.
Let R(n) be the set of squares that have vertices at integer coordinates and lie in the region of the plane |x|+|y| <= n+1, and let S(n) be the set of squares that have vertices at integer coordinates and lie in the region of the plane |x|+|y-1/2| <= n+2.  Further let T be the collection of rectangular tiles with dimensions i X 1 or 1 X i with i arbitrary.  Then a(2n) is the number of ways to tile R(n) using tiles from T and a(2n+1) is the number of ways to tile S(n) using tiles from T. (Note R(n) is the Aztec diamond of order n.) - Steve Butler, Jan 26 2015

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jin-Guo Liu, Table of n, a(n) for n = 0..39 (terms 1..33 from Robert Gerbicz, 34..35 from P. Butera and M. Pernici, 37..38 from Casey Mills Davis)
P. Butera and M. Pernici, Sums of permanental minors using Grassmann algebra, arXiv preprint arXiv:1406.5337 [hep-lat], 2014.
Casey Mills Davis, C++ program used to generate a(n) for n = 36..37
Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]
Xuanzhao Gao, Xiaofeng Li, and Jinguo Liu, Programming guide for solving constraint satisfaction problems with tensor networks, arXiv:2501.00227 [physics.comp-ph], 2024. See p. 24.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.372.
Jin-Guo Liu, Xun Gao, Madelyn Cain, Mikhail D. Lukin and Sheng-Tao Wang, Computing solution space properties of combinatorial optimization problems via generic tensor networks, arXiv:2205.03718 [cond-mat.stat-mech], 2022. Terms 38..39.
I. Mezo, Periodicity of the Last Digits of Some Combinatorial Sequences, J. Int. Seq. 17 (2014) #14.1.1.
B. D. Stosic, T. Stosic, I. P. Fittipaldi and J. J. P. Veerman, Residual entropy of the square Ising antiferromagnet in the maximum critical field: the Fibonacci matrix, Journal of Physics A: Mathematical and General, Volume 30, Number 10, 1997, pp. L331-L337.
Peter Tittmann, Enumeration in Graphs
Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hard Square Entropy Constant
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for sequences related to binary matrices

Cf. A027683 for toroidal version.
Table of values for n x m matrices: A089934.
Cf. A232833 for refinement by number of 1's.
Cf. also A191779.
Cf. A201511, A212270.

A006506 := proc(N) local i,j,p,q; p := 1+x11;
if n=0 then return 1 fi;
for i from 2 to N do
   q := p-select(has,p,x||(i-1)||1);
   p := p+expand(q*x||i||1)
od;
for j from 2 to N do
   q := p-select(has,p,x1||(j-1));
   p := subs(x1||(j-1)=1,p)+expand(q*x1||j);
   for i from 2 to N do
      q := p-select(has,p,{x||(i-1)||j,x||i||(j-1)});
      p := subs(x||i||(j-1)=1,p)+expand(q*x||i||j);
   od
od;
map(icontent,p)
end:
seq(A006506(n), n=0..15);

a[n_] := a[n] = (p = 1 + x[1, 1]; Do[q = p - Select[p, ! FreeQ[#, x[i-1, 1]] &]; p = p + Expand[q*x[i, 1]], {i, 2, n}]; Do[q = p - Select[p, ! FreeQ[#, x[1, j-1]] &]; p = (p /. x[i, j-1] :> 1) + Expand[q*x[1, j]]; Do[q = p - Select[ p, ! FreeQ[#, x[i-1, j]] || ! FreeQ[#, x[i, j-1]] &]; p = (p /. x[i, j-1] :> 1) + Expand[q*x[i, j]], {i, 2, n}], {j, 2, n}]; p /. x[, ] -> 1); a /@ Range[14] (* Jean-François Alcover, May 25 2011, after Maple prog. *)
Table[With[{g = GridGraph[{n, n}]}, Count[Subsets[Range[n^2], Length @ First @ FindIndependentVertexSet[g]], ?(IndependentVertexSetQ[g, #] &)]], {n, 5}] (* _Eric W. Weisstein, May 28 2017 *)

a(n)=L=fibonacci(n+2);p=v=vector(L,i,1);c=0; for(i=0,2^n-1,j=i;while(j&&j%4<3,j\=2);if(j%4<3,p[c++]=i)); for(i=2,n,w=vector(L,j,0); for(j=1,L, for(k=1,L,if(bitand(p[j],p[k])==0,w[j]+=v[k])));v=w); sum(i=1,L,v[i]) \\ Robert Gerbicz, Jun 17 2011

Limit_{n->oo} a(n)^(1/n^2) = c1 = 1.50304... is the hard square entropy constant A085850. - Benoit Cloitre, Nov 16 2003
a(n) appears to behave like A * c3^n * c1^(n^2) where c1 is as above, c3 = 1.143519129587 approximately, A = 1.0660826 approximately. This is based on numerical analysis of a(n) for n up to 19. - Brendan McKay, Nov 16 2003
From n up to 39 we have A = 1.06608266035... - Vaclav Kotesovec, Jan 29 2024

Sequence extended by Paul Zimmermann, Mar 15 1996
Maple program updated and sequence extended by Robert Israel, Jun 16 2011
a(0)=1 prepended by Alois P. Heinz, Jan 29 2024

A172225 Number of ways to place 2 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 2, 24, 96, 260, 570, 1092, 1904, 3096, 4770, 7040, 10032, 13884, 18746, 24780, 32160, 41072, 51714, 64296, 79040, 96180, 115962, 138644, 164496, 193800, 226850, 263952, 305424, 351596, 402810, 459420, 521792, 590304

Offset: 1

Vaclav Kotesovec, Jan 29 2010

A wazir is a (fairy chess) leaper [0,1].

Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p. 829.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A036464, A061995, A172132, A172123, A172137.

I:=[0, 2, 24, 96, 260]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[n*(n-1)*(n^2+n-4)/2: n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

Table[n (n - 1) (n^2 + n - 4) / 2, {n, 40}] (* Vincenzo Librandi, Apr 30 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,24,96,260},40] (* Harvey P. Dale, Jun 04 2023 *)

Explicit formula (Christian Poisson, 1990): a(n) = n*(n-1)*(n^2+n-4)/2.
G.f.: 2*x^2*(2*x^2-7*x-1)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Apr 30 2013
a(n) = 2*A239352(n). - R. J. Mathar, Jan 09 2018
a(n) = A232833(n,2). - R. J. Mathar, Apr 11 2024

A201511 Number of ways to place n nonattacking wazirs on an n X n board.

Original entry on oeis.org

1, 1, 2, 22, 405, 10741, 368868, 15516804, 771464278, 44218721793, 2868879752822, 207739939478618, 16602826428818482, 1451305771147909684, 137715836041691050398, 14096224186664736126206, 1547966111897855935957132, 181519663430661533452513680, 22636566614411901986006002896

Offset: 0

Vaclav Kotesovec, Dec 02 2011

Wazir is a leaper [0,1].

Vaclav Kotesovec, Table of n, a(n) for n = 0..21 (terms a(11)-a(18) computed by Peter Tittmann)
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 382.
Peter Tittmann, Polynomials of nxn Grid Graphs [broken link]

Cf. A172225, A172226, A172227, A172228, A178409, A201507, A201508, A201510.
Cf. A006506.
Main diagonal of A232833.

Asymptotics (Vaclav Kotesovec, Nov 29 2011): a(n) ~ n^(2n)/n!*exp(-5/2).

a(19)-a(20) from Vaclav Kotesovec, Aug 30 2016

a(0)=1 prepended by Alois P. Heinz, Apr 16 2024

A172226 Number of ways to place 3 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 22, 276, 1474, 5248, 14690, 35012, 74326, 144544, 262398, 450580, 739002, 1166176, 1780714, 2642948, 3826670, 5420992, 7532326, 10286484, 13830898, 18336960, 24002482, 31054276, 39750854, 50385248, 63287950

Offset: 1

Vaclav Kotesovec, Jan 29 2010

A wazir is a (fairy chess) leaper [0,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Eric Weisstein's World of Mathematics, Grid Graph
Wikipedia, Wazir (chess)
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A172225, A047659, A061996, A172124, A172134, A172138.

I:=[0, 0, 22, 276, 1474, 5248, 14690, 35012]; [n le 8 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[0] cat [(n-2)*(n^5+2*n^4-11*n^3-10*n^2+42*n-12)/6: n in [2..30]]; // Vincenzo Librandi, Apr 30 2013

A172226:=n->`if`(n=1, 0, (n-2)*(n^5 + 2*n^4 - 11*n^3 - 10*n^2 + 42*n - 12)/6); seq(A172226(n), n=1..60); # Wesley Ivan Hurt, Feb 06 2014

CoefficientList[Series[2 x^2 (x^5 - 9 x^4 + 22 x^3 - 2  x^2 - 61 x - 11) / (x-1)^7, {x, 0, 60}], x] (* Vincenzo Librandi, Apr 30 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,22,276,1474,5248,14690,35012},30] (* Harvey P. Dale, Apr 08 2022 *)

a(n) = (n-2)*(n^5 + 2*n^4 - 11*n^3 - 10*n^2 + 42*n - 12)/6, n>=2.
G.f.: 2*x^3*(x^5-9*x^4+22*x^3-2*x^2-61*x-11)/(x-1)^7. - Vaclav Kotesovec, Mar 25 2010
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Vincenzo Librandi, Apr 30 2013
a(n) = A232833(n,3). - R. J. Mathar, Apr 11 2024

A172227 Number of ways to place 4 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 6, 405, 5024, 31320, 133544, 446421, 1258590, 3126724, 7042930, 14669709, 28658436, 53069000, 93909924, 159819965, 262913874, 419816676, 652912510, 991835749, 1475233800, 2152832664, 3087838016, 4359706245, 6067321574, 8332617060, 11304678954

Offset: 1

Vaclav Kotesovec, Jan 29 2010

A wazir is a (fairy chess) leaper [0,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. Brazeal Slides on a Chessboard, Math Horizons, Vol. 27, pp. 24-27, April 2020.
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Eric Weisstein's World of Mathematics, Grid Graph
Wikipedia, Fairy chess piece
Wikipedia, Wazir (chess)

Cf. A172225, A172226, A061994, A061997, A172127, A172135, A172139, A006506.

CoefficientList[Series[- x^2 (4 x^8 - 26 x^7 + 3 x^6 + 303 x^5 - 736 x^4 + 180 x^3 + 1595 x^2 + 351 x + 6) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (n^8-30n^6+24n^5+323n^4-504n^3-1110n^2+2760n-1224)/24, n>=3.
G.f.: -x^3*(4*x^8-26*x^7+3*x^6+303*x^5-736*x^4+180*x^3+1595*x^2+351*x+6)/(x-1)^9. - Vaclav Kotesovec, Apr 29 2011
a(n) = A232833(n,4). - R. J. Mathar, Apr 11 2024

Corrected a(3) and g.f., Vaclav Kotesovec, Apr 29 2011

A172228 Number of ways to place 5 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 1, 304, 10741, 127960, 870589, 4197456, 16005187, 51439096, 145085447, 369074128, 863338777, 1883786680, 3875953561, 7583888944, 14206566327, 25617069208, 44663199283, 75572017136, 124485188701, 200156902936, 314851577749, 485484612496, 735056106571, 1094434774968

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Wazir is a (fairy chess) leaper [0,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Eric Weisstein's World of Mathematics, Grid Graph
Wikipedia, Fairy chess piece
Wikipedia, Wazir (chess)

Cf. A172225, A172226, A172227, A108792, A061998, A172129, A172136, A172140, A006506.

CoefficientList[Series[x^2 (6 x^11 - 26 x^10 - 93 x^9 + 527 x^8 + 490 x^7 - 6710 x^6 + 13630 x^5 - 3954 x^4 - 26364 x^3 - 7452 x^2 - 293 x - 1) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (n^10-50n^8+40n^7+995n^6-1560n^5-8890n^4+21080n^3+24264n^2-97440n+59520)/120, n>=4.
For any fixed value of k > 1, a(n) = n^(2k)/k! - 5/2/(k-2)!*n^(2k-2) + ...
G.f.: x^3 * (6*x^11 -26*x^10 -93*x^9 +527*x^8 +490*x^7 -6710*x^6 +13630*x^5 -3954*x^4 -26364*x^3 -7452*x^2 -293*x -1) / (x-1)^11. - Vaclav Kotesovec, Apr 29 2011
a(n) = A232833(n,5). - R. J. Mathar, Apr 11 2024

Corrected a(4) and g.f., Vaclav Kotesovec, Apr 29 2011.

More terms from Vincenzo Librandi, May 28 2013

A178409 Number of ways to place 6 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 0, 114, 14650, 368868, 4216498, 30222074, 158918030, 669582340, 2387463550, 7470004954, 21036576578, 54315955588, 130382565930, 294116445082, 628800849110, 1282821452132, 2511317339446, 4739431178170

Offset: 1

Vaclav Kotesovec, May 27 2010

Wazir is a (fairy chess) leaper [0,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013

Cf. A172225, A172226, A172227, A172228.

CoefficientList[Series[- 2 x^3 (4 x^13 - 17 x^12 + 3 x^11 - 469 x^10 + 4084 x^9 - 10233 x^8 - 3482 x^7 + 66494 x^6 - 125152 x^5 + 35457 x^4 + 265655 x^3 + 93655 x^2 + 6584 x + 57) / (x - 1)^13, {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)

Explicit formula: a(n) = 1/720 * (n^12 -75*n^10 +60*n^9 +2365*n^8 -3720*n^7 -38085*n^6 +89580*n^5 +292834*n^4 -984960*n^3 -552240*n^2 +4128960*n -3160800), n >= 5.
G.f.: -2*x^4 * (4*x^13 -17*x^12 +3*x^11 -469*x^10 +4084*x^9 -10233*x^8 -3482*x^7 +66494*x^6 -125152*x^5 +35457*x^4 +265655*x^3 +93655*x^2 +6584*x +57)/(x-1)^13.
a(n) = A232833(n,6). - R. J. Mathar, Apr 11 2024

A371967 Irregular triangle T(r,w) read by rows: number of ways of placing w non-attacking wazirs on a 3 X r board.

Original entry on oeis.org

1, 1, 3, 1, 1, 6, 8, 2, 1, 9, 24, 22, 6, 1, 1, 12, 49, 84, 61, 18, 2, 1, 15, 83, 215, 276, 174, 53, 9, 1, 1, 18, 126, 442, 840, 880, 504, 158, 28, 2, 1, 21, 178, 792, 2023, 3063, 2763, 1478, 472, 93, 12, 1, 1, 24, 239, 1292, 4176, 8406, 10692, 8604, 4374, 1416, 297, 38, 2, 1, 27, 309

Offset: 0

R. J. Mathar, Apr 14 2024

			The triangle starts with r>=0 rows and w>=0 wazirs as
  1 ;
  1 3 1 ;
  1 6 8 2  ;
  1 9 24 22 6 1 ;
  1 12 49 84 61 18 2  ;
  1 15 83 215 276 174 53 9 1 ;
  1 18 126 442 840 880 504 158 28 2  ;
  1 21 178 792 2023 3063 2763 1478 472 93 12 1 ;
  1 24 239 1292 4176 8406 10692 8604 4374 1416 297 38 2  ;
  1 27 309 1969 7731 19591 32716 36257 26674 13035 4264 945 142 15 1 ;
  ...

Alois P. Heinz, Rows r = 0..165 (first 17 rows from R. J. Mathar)
R. J. Mathar, Bivariate Generating Functions for Non-attacking Wazirs on Rectangular Boards
Jacob A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869 [math.CO] (2014) Table 2.
Wikipedia, Wazir (chess)

Cf. A051736 (row sums), A035607 (on 2Xr board), A011973 (on 1Xr board), A232833 (on rXr board).
T(n,n) gives A371978.
Row maxima give A371979.
Cf. A007494.

b:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*
      x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Apr 14 2024

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[If[BitAnd[j, l] > 0, 0, Expand[b[n - 1, j]*x^DigitCount[j, 2, 1]]], {j, {0, 1, 2, 4, 5}}]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 05 2024, after Alois P. Heinz *)

T(r,0) = 1.
T(r,1) = 3*r.
T(r,2) = A064225(r-1).
T(r,3) = A172229(r).
T(r,4) = 27*r^4/8 -117*r^3/4 +829*r^2/8 -715*r/4 +126. [Siehler Table 3]
T(3,w) = A232833(3,w).
G.f.: (1+x*y) *(1 +x*y +x*y^2 -x^2*y^3)/(1 -x -x*y -x^2*y^3 -2*x^2*y -3*x^2*y^2 -x^3*y^2 +x^3*y^4 +x^4*y^4). - R. J. Mathar, Apr 21 2024

A232569 Triangle T(n, k) = number of non-equivalent (mod D_4) binary n X n matrices with k pairwise not adjacent 1's; k=0,...,n^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 1, 3, 6, 6, 3, 1, 0, 0, 0, 0, 1, 3, 17, 40, 62, 45, 20, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 43, 210, 683, 1425, 1936, 1696, 977, 366, 101, 21, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 84, 681, 4015, 16149, 46472, 95838, 143657

Offset: 1

Heinrich Ludwig, Nov 29 2013

Also number of non-equivalent ways to place k non-attacking wazirs on an n X n board.
Two matrix elements are considered adjacent, if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).
Counted for this sequence are equivalence classes induced by the dihedral group D_4. If equivalent matrices are being destinguished, the corresponding numbers are A232833(n).
Row index starts from n = 1, column index k ranges from 0 to n^2.
T(n, 1) = A008805(n-1); T(n, 2) = A232567(n) for n >= 2; T(n, 3) = A232568(n) for n >= 2;
Into an n X n binary matrix there can be placed maximally A000982(n) = ceiling(n^2/2) pairwise not adjacent 1's.

			Triangle begins:
1,1;
1,1,1,0,0;
1,3,6,6,3,1,0,0,0,0;
1,3,17,40,62,45,20,4,1,0,0,0,0,0,0,0,0;
1,6,43,210,683,1425,1936,1696,977,366,101,21,5,1,0,0,0,0,0,0,0,0,0,0,0,0;
...
There are T(3, 2) = 6 non-equivalent binary 3 X 3 matrices with 2 not adjacent 1's (and no other 1's):
  [1 0 0]   [0 1 0]   [1 0 0]   [0 1 0]   [1 0 1]   [1 0 0]
  |0 0 0|   |0 0 0|   |0 1 0|   |1 0 0|   |0 0 0|   |0 0 1|
  [0 0 1]   [0 1 0]   [0 0 0]   [0 0 0]   [0 0 0]   [0 0 0]

Heinrich Ludwig, Rows n = 1..8 of irregular triangle, flattened

Cf. A232567, A232568, A239576, A008805, A000982, A201511, A232833.

cp's OEIS Frontend

A006506 Number of n X n binary matrices with no 2 adjacent 1's, or number of configurations of non-attacking princes on an n X n board, where a "prince" attacks the four adjacent (non-diagonal) squares. Also number of independent vertex sets in an n X n grid.

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A172225 Number of ways to place 2 nonattacking wazirs on an n X n board.

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A201511 Number of ways to place n nonattacking wazirs on an n X n board.

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A172226 Number of ways to place 3 nonattacking wazirs on an n X n board.

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A172227 Number of ways to place 4 nonattacking wazirs on an n X n board.

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A172228 Number of ways to place 5 nonattacking wazirs on an n X n board.

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A178409 Number of ways to place 6 nonattacking wazirs on an n X n board.

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