A172225 -id:A172225 - cp's OEIS frontend

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

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

A232833 Triangle read by rows: T(n,k) = number of n X n binary matrices with k pairwise nonadjacent 1's, n >= 0, k = 0..n^2.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 0, 0, 1, 9, 24, 22, 6, 1, 0, 0, 0, 0, 1, 16, 96, 276, 405, 304, 114, 20, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 25, 260, 1474, 5024, 10741, 14650, 12798, 7157, 2578, 618, 106, 14, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 36, 570, 5248, 31320, 127960, 368868

Offset: 0

Heinrich Ludwig, Dec 01 2013

Also number of 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).
If only non-equivalent (mod D_4) matrices are counted, the corresponding numbers are given by A232569.
Rows with trailing zeros dropped give the coefficients of the independence polynomial for the n X n grid graph. - Eric W. Weisstein, May 31 2017

			Triangle begins:
  1;
  1,  1;
  1,  4,  2,   0,   0;
  1,  9, 24,  22,   6,   1,   0,  0, 0, 0;
  1, 16, 96, 276, 405, 304, 114, 20, 2, 0, 0, 0, 0, 0, 0, 0, 0;
  ...

Alois P. Heinz, Rows n = 0..17, flattened (first 8.5 rows from Heinrich Ludwig)
R. J. Mathar, Bivariate generating functions for non-attacking Wazirs on rectangular boards, viXra:2404.0122 (2024) page 14
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Independence Polynomial

Cf. A232569, A006506 (row sums).

Main diagonal gives A201511.

b:= proc(n, l) option remember; local k;
      if n=0 then 1
    elif min(l)>0 then b(n-1, map(x-> x-1, l))
    else for k while l[k]>0 do od;
         b(n, subsop(k=1, l))+expand(x*`if`(n>0, `if`(k (p-> seq(coeff(p,x,i), i=0..n^2))(b(n, [0$n])):
seq(T(n), n=0..6);  # Alois P. Heinz, Apr 16 2024

b[n_, l_] := b[n, l] = Module[{k},
   Which[n == 0, 1,
   Min[l] > 0, b[n - 1, l - 1],
   True, For[k = 1, l[[k]] > 0, k++];
      b[n, ReplacePart[l, k -> 1]] + Expand[x*If[n > 0, If[k < Length[l],
      b[n, ReplacePart[l, {k -> 2, k + 1 -> 1}]],
      b[n, ReplacePart[l, k -> 2]], 0]]]]];
T[n_] := With[{p = b[n, Table[0, {n}]]}, Table[Coefficient[p, x, i], {i, 0, n^2}]]
Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Aug 09 2024, after Alois P. Heinz *)

T(n,0) = A000012(n);
T(n,1) = A000290(n), n >= 1;
T(n,2) = A172225(n), n >= 2;
T(n,3) = A172226(n), n >= 2;
T(n,4) = A172227(n), n >= 2;
T(n,5) = A172228(n), n >= 3;
T(n,6) = A178409(n), n >= 3;
T(n,7) = A201507(n), n >= 3;
T(n,8) = A201508(n), n >= 3;
T(n,9) = A201510(n), n >= 3;

T(0,0)=1 inserted 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

A232567 Number of non-equivalent binary n X n matrices with two nonadjacent 1's.

Original entry on oeis.org

0, 1, 6, 17, 43, 84, 159, 262, 426, 635, 940, 1311, 1821, 2422, 3213, 4124, 5284, 6597, 8226, 10045, 12255, 14696, 17611, 20802, 24558, 28639, 33384, 38507, 44401, 50730, 57945, 65656, 74376, 83657, 94078, 105129, 117459, 130492, 144951, 160190, 177010

Offset: 1

Heinrich Ludwig, Nov 26 2013

Also: Number of non-equivalent ways to place two 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).
This sequence counts equivalence classes induced by the dihedral group D_4. If equivalent matrices are distinguished, the number of matrices is A172225(n).

			There are a(3) = 6 non-equivalent 3 X 3 matrices with two nonadjacent 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, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1)

Cf. A232568, A232569, A239576, A201511, A172225.

x='x+O('x^99); concat(0, Vec(x*(1+x+x^2)*(1+3*x-x^2+x^3)/((1+x)^3*(1-x)^5))) \\ Altug Alkan, Mar 14 2018

a(n) = (n^4 + 2*n^2 - 4*n)/16 if n is even; a(n) = (n^4 + 4*n^2 - 8*n + 3)/16 if n is odd.

G.f.: x * (1 + x + x^2)*(1 + 3*x - x^2 + x^3) / ((1 + x)^3*(1 - x)^5). - Bruno Berselli, Nov 28 2013

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

A201236 Number of ways to place 2 non-attacking wazirs on an n X n toroidal board.

Original entry on oeis.org

0, 2, 18, 88, 250, 558, 1078, 1888, 3078, 4750, 7018, 10008, 13858, 18718, 24750, 32128, 41038, 51678, 64258, 79000, 96138, 115918, 138598, 164448, 193750, 226798, 263898, 305368, 351538, 402750, 459358, 521728, 590238, 665278, 747250, 836568, 933658, 1038958

Offset: 1

Vaclav Kotesovec, Nov 28 2011

nonn

A wazir is a leaper [0,1].

V. Kotesovec, Non-attacking chess pieces, 6ed, p.402

Cf. A172225, A201237, A201238, A201239, A201240, A201241, A201242.

a(n) = n^2*(n^2-5)/2, n>=3.

G.f.: 2*x^2 * (2*x^5 - 9*x^4 + 15*x^3 - 9*x^2 - 4*x - 1)/(x-1)^5.

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

A201507 Number of ways to place 7 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 0, 20, 12798, 763144, 15516804, 170842828, 1264750240, 7084450248, 32251861624, 125030824732, 426265242412, 1308045124808, 3675893768908, 9586626461484, 23445303141400, 54219244028296, 119372323892736, 251614892723068, 510130577706724, 998740710435208

Offset: 1

Vaclav Kotesovec, Dec 02 2011

nonn

Wazir is a leaper [0,1].

V. Kotesovec, Non-attacking chess pieces

Cf. A172225, A172226, A172227, A172228, A178409.

Explicit formula: n^14/5040 - n^12/48 + n^11/60 + 137*n^10/144 - 3*n^9/2 - 1139*n^8/48 + 223*n^7/4 + 59293*n^6/180 - 3191*n^5/3 - 8719*n^4/4 + 51737*n^3/5 + 10914*n^2/7 - 40708*n + 37228, n>=6.
G.f.: 2*x^4*(5*x^16 - 31*x^15 + 193*x^14 - 1683*x^13 + 5093*x^12 + 12431*x^11 - 111239*x^10 + 214181*x^9 + 187845*x^8 - 1518841*x^7 + 2546483*x^6 - 775465*x^5 - 6212549*x^4 - 2702167*x^3 - 286637*x^2 - 6249*x - 10)/(x-1)^15.
a(n) = A232833(n,7). - R. J. Mathar, Apr 11 2024

A201508 Number of ways to place 8 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 0, 2, 7157, 1143638, 44031035, 771464278, 8219304992, 62114308624, 364798895986, 1765597908290, 7329246973785, 26849172347850, 88645482921449, 268042562131202, 751635857876290, 1974215715426992, 4896315981217168, 11542835604897814, 26008912447737323

Offset: 1

Vaclav Kotesovec, Dec 02 2011

nonn

Wazir is a leaper [0,1].

V. Kotesovec, Non-attacking chess pieces

Cf. A172225, A172226, A172227, A172228, A178409, A201507.

Explicit formula: n^16/40320 - n^14/288 + n^13/360 + 623*n^12/2880 - 41*n^11/120 - 5521*n^10/720 + 649*n^9/36 + 941767*n^8/5760 - 12485*n^7/24 - 577177*n^6/288 + 3102289*n^5/360 + 12378183*n^4/1120 - 1545219*n^3/20 + 1588751*n^2/120 + 581605*n/2 - 308806, n>=7.
G.f.: -x^4*(12*x^19 - 122*x^18 + 1130*x^17 - 6776*x^16 + 11180*x^15 + 33894*x^14 + 82772*x^13 - 1938093*x^12 + 7575029*x^11 - 10487057*x^10 - 11993287*x^9 + 70715064*x^8 - 109013258*x^7 + 41757444*x^6 + 331980470*x^5 + 173609451*x^4 + 25561181*x^3 + 1022241*x^2 + 7123*x + 2)/(x-1)^17.
a(n) = A232833(n,8). - R. J. Mathar, Apr 11 2024

cp's OEIS Frontend

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

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A232833 Triangle read by rows: T(n,k) = number of n X n binary matrices with k pairwise nonadjacent 1's, n >= 0, k = 0..n^2.

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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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A232567 Number of non-equivalent binary n X n matrices with two nonadjacent 1's.

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

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A201236 Number of ways to place 2 non-attacking wazirs on an n X n toroidal board.

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

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