A172137 -id:A172137 - 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

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)

A172138 Number of ways to place 3 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 4, 84, 452, 1772, 5596, 14888, 34640, 72712, 140716, 255036, 437968, 718980, 1136092, 1737376, 2582576, 3744848, 5312620, 7391572, 10106736, 13604716, 18056028, 23657560, 30635152, 39246296, 49782956, 62574508, 77990800

Offset: 1

Vaclav Kotesovec, Jan 26 2010

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

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, A172134, A172137.

[0,4,84,452,1772] cat [(n^6 -27*n^4 +120*n^3 +74*n^2 -1608*n +2976)/6: n in [6..50]]; // G. C. Greubel, Apr 19 2022

CoefficientList[Series[4x(1+14*x-13*x^2+58*x^3-29*x^4-9*x^5+x^6+ 33*x^7- 45*x^8 +23*x^9-4*x^10)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,4,84,452,1772,5596,14888,34640,72712,140716,255036,437968},30] (* Harvey P. Dale, Mar 11 2023 *)

[0,4,84,452,1772]+[(n^6 -27*n^4 +120*n^3 +74*n^2 -1608*n +2976)/6 for n in (6..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^6 - 27*n^4 + 120*n^3 + 74*n^2 - 1608*n + 2976)/6, n >=6.

G.f.: 4*x^2*(1 + 14*x - 13*x^2 + 58*x^3 - 29*x^4 - 9*x^5 + x^6 + 33*x^7 - 45*x^8 + 23*x^9 - 4*x^10)/(1-x)^7. - Vaclav Kotesovec, Mar 25 2010

A172139 Number of ways to place 4 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 1, 126, 1168, 7334, 35749, 137970, 438984, 1208246, 2969389, 6662480, 13873100, 27144408, 50389581, 89424014, 152638280, 251834530, 403250693, 628798516, 957543164, 1427453780, 2087456085, 2999819778, 4242915176

Offset: 1

Vaclav Kotesovec, Jan 26 2010

nonn

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

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, A172135, A172137, A172138.

CoefficientList[Series[x(1+117*x+70*x^2+1274*x^3+1333*x^4-2109*x^5-462*x^6 +8858*x^7-17006*x^8+15166*x^9-6838*x^10+1478*x^11-650*x^12+760*x^13-376*x^14 +64*x^15)/(1-x)^9, {x,0,40}], x] (* Vincenzo Librandi, May 27 2013 *)

[0,1,126,1168,7334,35749,137970,438984] + [(n^8 -54*n^6 +240*n^5 +827*n^4 -8592*n^3 +10362*n^2 +75600*n -204864)/24 for n in (9..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^8 - 54*n^6 + 240*n^5 + 827*n^4 - 8592*n^3 + 10362*n^2 + 75600*n - 204864)/24, n >= 9.

G.f.: x^2*(1 + 117*x + 70*x^2 + 1274*x^3 + 1333*x^4 - 2109*x^5 - 462*x^6 + 8858*x^7 - 17006*x^8 + 15166*x^9 - 6838*x^10 + 1478*x^11 - 650*x^12 + 760*x^13 - 376*x^14 + 64*x^15)/(1-x)^9. - Vaclav Kotesovec, Mar 25 2010

A172140 Number of ways to place 5 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 0, 126, 2032, 20502, 160696, 929880, 4117520, 15037036, 47368960, 132577826, 336828368, 789558314, 1729320120, 3574328936, 7027309888, 13226773092, 23959787480, 41954706558, 71276149776, 117848892710, 190142197976

Offset: 1

Vaclav Kotesovec, Jan 26 2010

nonn

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

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, A172136, A172137, A172138, A172139.

CoefficientList[Series[2x^2(100x^19 -648x^18 +1450x^17 -2126x^16 +10452x^15 - 43872x^14 +92798x^13 -100834x^12 +56460x^11 -61636x^10 +182288x^9 -303224x^8 + 275038x^7 -128982x^6 +21681x^5 +1933x^4 -13072x^3 -2540x^2 -323x-63)/(x-1)^11, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)

[0,0,126,2032,20502,160696,929880,4117520,15037036,47368960,132577826] + [(n^10 -90*n^8 +400*n^7 +2915*n^6 -26880*n^5 +2430*n^4 +609920*n^3 - 1517496*n^2 -4188480*n +16581120)/120 for n in (12..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^10 - 90*n^8 + 400*n^7 + 2915*n^6 - 26880*n^5 + 2430*n^4 + 609920*n^3 - 1517496*n^2 - 4188480*n + 16581120)/120, n >= 12.
For any fixed value of k > 1, a(n) = n^(2k) /k! - 9n^(2k - 2) /2/(k - 2)! + 20n^(2k - 3) /(k - 2)! + ...
G.f.: 2*x^3 * (100*x^19 -648*x^18 +1450*x^17 -2126*x^16 +10452*x^15 -43872*x^14 +92798*x^13 -100834*x^12 +56460*x^11 -61636*x^10 +182288*x^9 -303224*x^8 +275038*x^7 -128982*x^6 +21681*x^5 +1933*x^4 -13072*x^3 -2540*x^2 -323*x -63) / (x-1)^11. - Vaclav Kotesovec, Mar 25 2010

A244284 Number of ways to place n nonattacking zebras on an n X n chessboard.

Original entry on oeis.org

1, 6, 84, 1168, 20502, 525796, 18939708, 802444170, 38934305898, 2170312156170

Offset: 1

Vaclav Kotesovec, Jun 25 2014

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

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.363
Wikipedia, Fairy chess piece

Cf. A172137, A172138, A172139, A172140.

Cf. A201540, A201511, A201861, A201513.

a(n) ~ n^(2*n)/n! * exp(-9/2).

cp's OEIS Frontend

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

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

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A172138 Number of ways to place 3 nonattacking zebras on an n X n board.

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A172139 Number of ways to place 4 nonattacking zebras on an n X n board.

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A172140 Number of ways to place 5 nonattacking zebras on an n X n board.

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A244284 Number of ways to place n nonattacking zebras on an n X n chessboard.

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