A172132 -id:A172132 - cp's OEIS frontend

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

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

A172134 Number of ways to place 3 nonattacking knights on an n X n board.

Original entry on oeis.org

0, 4, 36, 276, 1360, 4752, 13340, 32084, 68796, 135040, 247152, 427380, 705144, 1118416, 1715220, 2555252, 3711620, 5272704, 7344136, 10050900, 13539552, 17980560, 23570764, 30535956, 39133580, 49655552, 62431200, 77830324

Offset: 1

Vaclav Kotesovec, Jan 26 2010

E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav 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 (7,-21,35,-35,21,-7,1).

Cf. A047659, A172124, A172132.

Column k=3 of A244081.

[n le 3 select (n*(n-1))^2 else (n-2)*(n+5)*(n^4 -3*n^3 -8*n^2 +66*n -108)/6: n in [1..50]]; // G. C. Greubel, Apr 18 2022

CoefficientList[Series[4x(3x^8 -20x^7 +43x^6 -38x^5 +23x^4 -11x^3 -27x^2 -2x -1)/ (x-1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)

def A172134(n):
    if (n<4): return (n*(n-1))^2
    else: return (n-2)*(n+5)*(n^4 -3*n^3 -8*n^2 +66*n -108)/6
[A172134(n) for n in (1..50)] # G. C. Greubel, Apr 18 2022

Explicit formula (Karl Fabel, 1966): a(n) = (n - 2)*(n + 5)*(n^4 - 3*n^3 - 8*n^2 + 66*n - 108)/6, for n >= 4.
G.f.: 4*x^2*(3*x^8-20*x^7+43*x^6-38*x^5+23*x^4-11*x^3-27*x^2-2*x-1)/(x-1)^7. - Vaclav Kotesovec, Mar 25 2010
From G. C. Greubel, Apr 18 2022: (Start)
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), for n >= 11.
E.g.f.: (1/6)*(-1080 - 312*x + 12*x^2 +13*x^3 + (1080 - 768*x + 228*x^2 + 38*x^4 + 15*x^5 + x^6)*exp(x)). (End)

A172135 Number of ways to place 4 nonattacking knights on an n X n board.

Original entry on oeis.org

0, 1, 18, 412, 4436, 26133, 111066, 376560, 1080942, 2732909, 6253408, 13204356, 26100160, 48819677, 87137934, 149398608, 247349946, 397168485, 620696612, 946921684, 1413726108, 2069939461, 2977725410, 4215337872

Offset: 1

Vaclav Kotesovec, Jan 26 2010

E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav 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 (9,-36,84,-126,126,-84,36,-9,1).

Cf. A061994, A172127, A172132, A172134.

Column k=4 of A244081.

[0,1,18,412,4436] cat [(n^8 -54*n^6 +144*n^5 +1019*n^4 -5232*n^3 -2022*n^2 +51120*n -77184)/24: n in [6..50]]; // G. C. Greubel, Apr 19 2022

CoefficientList[Series[x*(1 +9*x +286*x^2 +1292*x^3 -345*x^4 +3099*x^5 -5142*x^6 +3606*x^7 -1162*x^8 -390*x^9 +690*x^10 -312*x^11 +48*x^12)/(1-x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, May 26 2013 *)

[0,1,18,412,4436] + [(n^8 -54*n^6 +144*n^5 +1019*n^4 -5232*n^3 -2022*n^2 +51120*n -77184)/24 for n in (6..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^8 - 54*n^6 + 144*n^5 + 1019*n^4 - 5232*n^3 - 2022*n^2 + 51120*n - 77184)/24, n >= 6. (Karl Fabel, 1966)
G.f.: x^2 * ( 1 + 9*x + 286*x^2 + 1292*x^3 - 345*x^4 +3099*x^5 - 5142*x^6 + 3606*x^7 - 1162*x^8 - 390*x^9 + 690*x^10 - 312*x^11 + 48*x^12) / (1-x)^9. - Vaclav Kotesovec, Mar 25 2010
E.g.f.: x^2/2! + 18*x^3/3! + 412*x^4/4! + 4436*x^5/5! + (1/120)*(385920 + 161040*x + 17940*x^2 - 1200*x^3 - 2660*x^4 - 4484*x^5 + (-385920 + 224880*x - 49860*x^2 + 2940*x^3 + 3250*x^4 + 1920*x^5 + 1060*x^6 + 140*x^7 + 5*x^8)*exp(x)). - G. C. Greubel, Apr 19 2022

A172136 Number of ways to place 5 nonattacking knights on an n X n board.

Original entry on oeis.org

0, 0, 2, 340, 9386, 97580, 649476, 3184708, 12472084, 41199404, 119171110, 309957412, 739123094, 1639655452, 3422020324, 6778432292, 12833460256, 23356032940, 41051290730, 69954580804

Offset: 1

Vaclav Kotesovec, Jan 26 2010

For any fixed value of k>1, a(n) = n^(2*k) /k! - 9*n^(2*k - 2) /2/(k - 2)! + 12*n^(2*k - 3) /(k - 2)! + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav 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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A108792, A172129, A172132, A172134, A172135.

Column k=5 of A244081.

CoefficientList[Series[2x^2(74 x^15 -518x^14 +1110x^13 +1046x^12 -11332x^11 + 29950x^10 -42430x^9 +32476x^8 -11684x^7 -1000x^6 +15021x^5 -18443x^4 -6352x^3 - 2878x^2 -159x -1)/(x-1)^11, {x,0,40}], x] (* Vincenzo Librandi, May 02 2013 *)

[0,0,2,340,9386,97580,649476] + [(n^10 -90*n^8 +240*n^7 +3235*n^6 - 16320*n^5 -40530*n^4 +396480*n^3 -231656*n^2 -3359520*n +6509280)/120 for n in (8..50)] # G. C. Greubel, Apr 19 2022

Explicit formula: a(n) = (n^10 - 90*n^8 + 240*n^7 + 3235*n^6 - 16320*n^5 - 40530*n^4 + 396480*n^3 - 231656*n^2 - 3359520*n + 6509280)/120, n >= 8.

G.f.: 2*x^3 * (74*x^15 -518*x^14 +1110*x^13 +1046*x^12 -11332*x^11 +29950*x^10 -42430*x^9 +32476*x^8 -11684*x^7 -1000*x^6 +15021*x^5 -18443*x^4 -6352*x^3 -2878*x^2 -159*x -1) / (x-1)^11. [Vaclav Kotesovec, Mar 25 2010]

A244081 Number T(n,k) of ways to place k nonattacking knights on an n X n board; triangle T(n,k), n>=0, 0<=k<=A030978(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 4, 1, 1, 9, 28, 36, 18, 2, 1, 16, 96, 276, 412, 340, 170, 48, 6, 1, 25, 252, 1360, 4436, 9386, 13384, 12996, 8526, 3679, 994, 158, 15, 1, 1, 36, 550, 4752, 26133, 97580, 257318, 491140, 688946, 716804, 556274, 323476, 141969, 47684, 12488, 2560, 393, 40, 2

Offset: 0

Alois P. Heinz, Jun 19 2014

In other words, the n-th row gives the coefficients of the independence polynomial of the n X n knight graph. - Eric W. Weisstein, May 05 2017

			T(4,8) = 6:
  ._______. ._______. ._______. ._______. ._______. ._______.
  |_|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|_|o| |o|_|o|_| |_|_|_|_| |o|_|_|o| |o|_|_|_| |_|_|_|o|
  |o|_|o|_| |_|o|_|o| |o|o|o|o| |o|_|_|o| |o|o|_|o| |o|_|o|o| .
.
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   6,    4,    1;
  1,  9,  28,   36,   18,    2;
  1, 16,  96,  276,  412,  340,   170,    48,    6;
  1, 25, 252, 1360, 4436, 9386, 13384, 12996, 8526, 3679, 994, 158, 15, 1;
  ...
As independence polynomials:
  1
  1 + x
  1 + 4*x + 6*x^2 + 4*x^3 + x^4
  1 + 9*x + 28*x^2 + 36*x^3 + 18*x^4 + 2*x^5
  1 + 16*x + 96*x^2 + 276*x^3 + 412*x^4 + 340*x^5 + 170*x^6 + 48*x^7 + 6*x^8
  ...

Alois P. Heinz, Rows n = 0..12, flattened
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Knight Graph
Eric Weisstein's World of Mathematics, Knights Problem

Columns k=0-6 give: A000012, A000290, A172132, A172134, A172135, A172136, A178499.
T(n,n) gives A201540.
Row sums give A141243.
Cf. A030978.

b:= proc(n, l) option remember; local d, f, g, k;
      d:= nops(l)/3; f:=false;
      if n=0 then 1
    elif l[1..d]=[f$d] then b(n-1, [l[d+1..3*d][], true$d])
    else for k while not l[k] do od; g:= subsop(k=f, l);
         if k>1 then g:=subsop(2*d-1+k=f, g) fi;
         if k2 then g:=subsop(  d-2+k=f, g) fi;
         if k(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, [true$(n*3)])):
seq(T(n), n=0..7);

b[n_, l_] := b[n, l] = Module[{d, f, g, k}, d = Length[l]/3; f = False; Which[n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n - 1, Join[l[[d + 1 ;; 3*d]], Array[True&, d]]], True, For[k = 1, !l[[k]], k++]; g = ReplacePart[l, k -> f];
     If [k > 1, g = ReplacePart[g, 2*d - 1 + k -> f]];
     If [k < d, g = ReplacePart[g, 2*d + 1 + k -> f]];
     If [k > 2, g = ReplacePart[ g, d - 2 + k -> f]];
     If [k < d - 1, g = ReplacePart[g, d + 2 + k -> f]];
     Expand[b[n, ReplacePart[l, k -> f]] + b[n, g]*x]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][
  b[n, Array[True&, n*3]]];
Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 28 2016, after Alois P. Heinz *)
Table[Count[IndependentVertexSetQ[KnightTourGraph[n, n], #] & /@ Subsets[Range[n^2], {k}], True], {n, 4}, {k, 0, If[n == 2, 4, (1 - (-1)^n + 2 n^2)/4]}] // Flatten (* Eric W. Weisstein, May 05 2017 *)

A172141 Number of ways to place 2 nonattacking nightriders on an n X n board.

Original entry on oeis.org

0, 6, 28, 96, 240, 518, 980, 1712, 2784, 4310, 6380, 9136, 12688, 17206, 22820, 29728, 38080, 48102, 59964, 73920, 90160, 108966, 130548, 155216, 183200, 214838, 250380, 290192, 334544, 383830, 438340, 498496, 564608, 637126

Offset: 1

Vaclav Kotesovec, Jan 26 2010

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Vaclav 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 (3,-1,-5,5,1,-3,1).

Cf. A036464, A172123, A172132, A172137.

[(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)): n in [1..40]]; // G. C. Greubel, Apr 21 2022

CoefficientList[Series[2*x*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2), {x,0,40}], x] (* Vincenzo Librandi, May 27 2013 *)

[(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)) for n in (1..40)] # G. C. Greubel, Apr 21 2022

Explicit formula (Christian Poisson, 1990): a(n) = n*(3*n^3 - 5*n^2 + 9*n - 4)/6 if n is even and a(n) = n*(n - 1)*(3*n^2 - 2*n + 7)/6 if n is odd.
G.f.: 2*x^2*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2). - Vaclav Kotesovec, Mar 25 2010
From G. C. Greubel, Apr 21 2022: (Start)
a(n) = (1/12)*n*(3*(-1)^n - (11 - 18*n + 10*n^2 - 6*n^3)).
E.g.f.: (x/12)*(-3*exp(-x) + (3 + 30*x + 26*x^2 + 6*x^3)exp(x)). (End)

A201540 Number of ways to place n nonattacking knights on an n X n board.

Original entry on oeis.org

1, 6, 36, 412, 9386, 257318, 8891854, 379978716, 19206532478, 1120204619108, 74113608972922, 5483225594409823, 448414229054798028, 40154319792412218900, 3906519894750904583838

Offset: 1

Vaclav Kotesovec, Dec 02 2011

a(n) = A244081(n,n). - Alois P. Heinz, Jun 19 2014

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 293.

Cf. A172132, A172134, A172135, A172136, A178499, A244081.

Cf. A244284, A201511, A201861, A201513, A141243.

b[n_, l_] := b[n, l] = Module[{d, f, g, k}, d = Length[l]/3; f = False; Which[n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n - 1, Join[l[[d + 1 ;; 3*d]], Array[True&, d]]], True, For[k = 1, ! l[[k]], k++]; g = ReplacePart[l, k -> f];
     If[k > 1, g = ReplacePart[g, 2*d - 1 + k -> f]];
     If[k < d, g = ReplacePart[g, 2*d + 1 + k -> f]];
     If[k > 2, g = ReplacePart[g, d - 2 + k -> f]];
     If[k < d - 1, g = ReplacePart[g, d + 2 + k -> f]];
     Expand[b[n, ReplacePart[l, k -> f]] + b[n, g]*x]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Array[True&, n*3]]];
a[n_] := T[n][[n + 1]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Mar 29 2016, after Alois P. Heinz's code for A244081 *)

a(n) ~ n^(2n)/n!*exp(-9/2). - Vaclav Kotesovec, Nov 29 2011

a(11) from Alois P. Heinz, Jun 19 2014
a(12)-a(13) from Vaclav Kotesovec, Jun 21 2014
a(14) from Vaclav Kotesovec, Aug 26 2016
a(15) from Vaclav Kotesovec, May 26 2021

A172137 Number of ways to place 2 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 6, 36, 112, 276, 582, 1096, 1896, 3072, 4726, 6972, 9936, 13756, 18582, 24576, 31912, 40776, 51366, 63892, 78576, 95652, 115366, 137976, 163752, 192976, 225942, 262956, 304336, 350412, 401526, 458032, 520296, 588696, 663622, 745476, 834672, 931636, 1036806

Offset: 1

Vaclav Kotesovec, Jan 26 2010

Zebra is a (fairy chess) leaper [2,3].

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav 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, A172123, A172132.

[n eq 1 select 0 else (n^4 -9*n^2 +40*n -48)/2: n in [1..50]]; // G. C. Greubel, Apr 19 2022

CoefficientList[Series[2x(3+3*x-4*x^2+8*x^3-4*x^4)/(1-x)^55, {x, 0, 40}], x] (* Vincenzo Librandi, May 26 2013 *)

[(n^4 -9*n^2 +40*n -48 +16*bool(n==1))/2 for n in (1..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^4 - 9*n^2 + 40*n - 48)/2, n >= 2. (Christian Poisson, 1990)
G.f.: 2*x^2*(3+3*x-4*x^2+8*x^3-4*x^4)/(1-x)^5. - Vaclav Kotesovec, Mar 25 2010
E.g.f.: (1/2)*(16*(3+x) + (-48 + 32*x - 2*x^2 + 6*x^3 + x^4)*exp(x)). - G. C. Greubel, Apr 19 2022

A172529 Number of ways to place 2 nonattacking knights on an n X n toroidal board.

Original entry on oeis.org

0, 2, 18, 88, 200, 486, 980, 1760, 2916, 4550, 6776, 9720, 13520, 18326, 24300, 31616, 40460, 51030, 63536, 78200, 95256, 114950, 137540, 163296, 192500, 225446, 262440, 303800, 349856, 400950, 457436, 519680, 588060, 662966, 744800, 833976, 930920, 1036070

Offset: 1

Vaclav Kotesovec, Feb 06 2010

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

Cf. A172132, A172517.

CoefficientList[Series[2 x (16 x^7 - 71 x^6 + 121 x^5 - 98 x^4 + 40 x^3 - 9 x^2 - 4 x - 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

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

G.f.: 2*x^2*(16*x^7-71*x^6+121*x^5-98*x^4+40*x^3-9*x^2-4*x-1)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010

More terms from Vincenzo Librandi, May 29 2013

A172964 Number of ways to place 2 nonattacking knights on an n X n cylindrical board.

Original entry on oeis.org

0, 4, 18, 92, 230, 522, 1022, 1808, 2970, 4610, 6842, 9792, 13598, 18410, 24390, 31712, 40562, 51138, 63650, 78320, 95382, 115082, 137678, 163440, 192650, 225602, 262602, 303968, 350030, 401130, 457622, 519872, 588258, 663170, 745010, 834192, 931142, 1036298, 1150110

Offset: 1

Vaclav Kotesovec, Feb 06 2010

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

Cf. A172529, A172132.

CoefficientList[Series[2 x (6 x^7 - 30 x^6 + 61 x^5 - 66 x^4 + 45 * x^3 - 21 x^2 + x - 2) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

a(n) = n*(n^3 - 9*n + 12)/2, n>=5.

G.f.: 2*x^2*(6*x^7-30*x^6+61*x^5-66*x^4+45*x^3-21*x^2+x-2)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010

More terms from Vincenzo Librandi, May 29 2013

cp's OEIS Frontend

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

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A172134 Number of ways to place 3 nonattacking knights on an n X n board.

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A172135 Number of ways to place 4 nonattacking knights on an n X n board.

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A172136 Number of ways to place 5 nonattacking knights on an n X n board.

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A244081 Number T(n,k) of ways to place k nonattacking knights on an n X n board; triangle T(n,k), n>=0, 0<=k<=A030978(n), read by rows.

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A172141 Number of ways to place 2 nonattacking nightriders on an n X n board.

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A201540 Number of ways to place n nonattacking knights on an n X n board.

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