A172138 -id:A172138 - cp's OEIS frontend

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

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

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

A172221 Number of ways to place 3 nonattacking zebras on a 3 X n board.

Original entry on oeis.org

1, 20, 84, 200, 403, 720, 1180, 1808, 2631, 3676, 4970, 6540, 8413, 10616, 13176, 16120, 19475, 23268, 27526, 32276, 37545, 43360, 49748, 56736, 64351, 72620, 81570, 91228, 101621, 112776, 124720, 137480, 151083, 165556, 180926, 197220

Offset: 1

Vaclav Kotesovec, Jan 29 2010

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

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, Zebra Graph
Wikipedia, Zebra (chess)

Cf. A172138, A061989.

CoefficientList[Series[(2 x^8 - 4 x^7 + 2 x^6 - 8 x^5 + 28 x^4 - 20 x^3 + 10 x^2 + 16 x + 1) / (x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (9*n^3 - 21*n^2 + 50*n - 48)/2, n>=6.

G.f.: x*(2*x^8-4*x^7+2*x^6-8*x^5+28*x^4-20*x^3+10*x^2+16*x+1)/(x-1)^4. - Vaclav Kotesovec, Mar 25 2010

More terms from Vincenzo Librandi, May 28 2013

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

A172226 Number of ways to place 3 nonattacking wazirs 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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A172221 Number of ways to place 3 nonattacking zebras on a 3 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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