A172200 -id:A172200 - cp's OEIS frontend

A225553 Longest checkmate in king and amazon versus king endgame on an n X n chessboard.

0, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21

Offset: 3

Vaclav Kotesovec, May 10 2013

nonn

An amazon (superqueen) moves like a queen and a knight.

			Longest win on an 8x8 chessboard: Ka1 AMb1 - Kd4, 1.AMb1-f5! Kd4-c4! 2.Ka1-b1 Kc4-b4! 3.Kb1-b2 Kb4-a4 4.AMf5-c5#, therefore a(8) = 4.

V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001)
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p.364 (2013), p. 544 (second edition, 2017).

Cf. A172200, A051223, A178967, A225551, A225552, A225554, A225555, A225556, A225557, A227437.

Conjecture: for n > 10, a(n) = floor((n+2)/2).

Empirical g.f.: -x^4*(x^9-x^8+x^4-x-1) / ((x-1)^2*(x+1)). - Colin Barker, May 11 2013

A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 48, 424, 1976, 6616, 17852, 41544, 86660, 166288, 298616, 508200, 827168, 1296744, 1968676, 2907016, 4189772, 5910944, 8182400, 11136168, 14926536, 19732600, 25760588, 33246664, 42459476, 53703216, 67320392, 83695144

Offset: 1

Vaclav Kotesovec, Jan 29 2010

An amazon (superqueen) moves like a queen and a knight.

Panos Louridas, idee & form 93/2007, pp. 2936-2938.

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,-8,0,14,-14,0,8,-5,1).

Cf. A047659, A051223, A051224, A061989, A172200.

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( 4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7 ))); // G. C. Greubel, Apr 29 2022

CoefficientList[Series[4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7), {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)

[(1/24)*(4*n^6 -40*n^5 +62*n^4 +628*n^3 -2904*n^2 +4074*n -1341 +3*(-1)^n*(2*n-1)) -4*(5*bool(n==1) +2*bool(n==2) -bool(n==3)) for n in (1..40)] # G. C. Greubel, Apr 29 2022

Explicit formula (Panos Louridas, 2007): a(n) = (2*n^6 - 20*n^5 + 31*n^4 + 314*n^3 - 1452*n^2 + 2040*n - 672)/12 if n is even (n >= 4) and a(n) = (2*n^6 - 20*n^5 + 31*n^4 + 314*n^3 - 1452*n^2 + 2034*n - 669)/12 if n is odd (n >= 5).
G.f.: 4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7). - Vaclav Kotesovec, Mar 24 2010
a(n) = (1/24)*(4*n^6 - 40*n^5 + 62*n^4 + 628*n^3 - 2904*n^2 + 4074*n - 1341 + 3*(-1)^n*(2*n-1)) - 20*[n=1] - 8*[n=2] + 4*[n=3]. - G. C. Greubel, Apr 29 2022

A173214 Number of ways to place 4 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 2, 112, 1754, 13074, 63400, 234014, 712248, 1882132, 4457246, 9679760, 19584514, 37367934, 67849336, 118085614, 198107620, 321870956, 508359070, 782972820, 1179105738, 1740089734, 2521359260, 3593085246, 5043058972

Offset: 1

Vaclav Kotesovec, Feb 12 2010

A amazon (superqueen) moves like a queen and a knight.

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
Wikipedia, Amazon (chess)

Cf. A172200, A172201, A061994.

CoefficientList[Series[2 x^4 (28 x^17 - 18 x^16 - 162 x^15 - 139 x^14 + 261 x^13 + 1268 x^12 + 2387 x^11 + 1220 x^10 - 5937 x^9 - 18637 x^8 - 30086 x^7 - 31557 x^6 - 23251 x^5 - 11716 x^4 - 3859 x^3 - 708 x^2 - 53 x - 1) / ((x + 1)^4 (x - 1)^9 (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

a(n) = n^8/24-5n^7/6+47n^6/9+43n^5/10-5053n^4/24+112585n^3/108-15433n^2/8+55669n/270+119917/54 + (n^3/4-21n^2/8+7n-3/2)*(-1)^n + 32/27*(n-1)*cos(2*Pi*n/3) + 40*sqrt(3)*sin(2*Pi*n/3)/81, n>=6.
Recurrence: a(n) = 3a(n-1)+a(n-2)-9a(n-3)+12a(n-5)+7a(n-6)-15a(n-7)-16a(n-8)+16a(n-9)+15a(n-10)-7a(n-11)-12a(n-12)+9a(n-14)-a(n-15)-3a(n-16)+a(n-17), n>=23. - Vaclav Kotesovec, Feb 18 2010
G.f.: 2x^5*(28x^17-18x^16-162x^15-139x^14+261x^13+1268x^12+2387x^11+1220x^10-5937x^9-18637x^8-30086x^7-31557x^6-23251x^5-11716x^4-3859x^3-708x^2-53x-1)/((x+1)^4*(x-1)^9*(x^2+x+1)^2). - Vaclav Kotesovec, Mar 24 2010

A174642 Number of ways to place 4 nonattacking amazons (superqueens) on a 4 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 12, 60, 180, 432, 900, 1692, 2940, 4800, 7452, 11100, 15972, 22320, 30420, 40572, 53100, 68352, 86700, 108540, 134292, 164400, 199332, 239580, 285660, 338112, 397500, 464412, 539460, 623280, 716532, 819900, 934092, 1059840

Offset: 1

Vaclav Kotesovec, Mar 25 2010

An amazon (superqueen) moves like a queen and a knight

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
The Oprisch Family Web Site
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A173214, A172201, A172200, A051223, A051224, A036464.

CoefficientList[Series[- 12 x^7 (x^3 + 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

G.f.: -12*x^8*(x^3+1)/(x-1)^5.

Explicit formula: a(n) = (n-7)(n^3-21n^2+158n-420), n>=7.

More terms from Vincenzo Librandi, May 30 2013

A178972 Number of ways to place 2 nonattacking amazons (superqueens) on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 144, 392, 896, 1620, 2800, 4356, 6624, 9464, 13328, 18000, 24064, 31212, 40176, 50540, 63200, 77616, 94864, 114264, 137088, 162500, 191984, 224532, 261856, 302760, 349200, 399776, 456704, 518364, 587248, 661500, 743904, 832352, 929936, 1034280, 1148800

Offset: 1

Vaclav Kotesovec, Jan 02 2011

An amazon (superqueen) moves like a queen and a knight.

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

Cf. A172200, A172517.

CoefficientList[Series[4 x^5 (8 x^6 - 7 x^5 - 30 x^4 + 23 x^3 + 44 x^2 - 26 x - 36) / ((x - 1)^5 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)

a(n) = 1/2*n^2*(n^2 -4*n -9/2 +(-1)^n/2), n>=5.

G.f.: 4*x^6*(8*x^6 -7*x^5 -30*x^4 +23*x^3 +44*x^2 -26*x -36)/((x-1)^5*(x+1)^3).

A178967 Number of ways to place 5 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 248, 7320, 82758, 562384, 2756122, 10771928, 35504296, 102677536, 267284836, 638673432, 1420555842, 2974232240, 5911536526, 11232560320, 20516606128, 36191817440, 61893239340, 102950022616, 167010533830, 264869097472, 411497661102, 627378473416, 940130628920, 1386570370640, 2015178519904, 2889176379864, 4090150245318, 5722507236712, 7918655437366, 10845295301648, 14710646654420, 19773136732920, 26351274869008, 34835414789584

Offset: 1

Vaclav Kotesovec, Jan 01 2011

An amazon (superqueen) moves like a queen and a knight.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
Wikipedia, Amazon (chess)

Cf. A173214, A172200, A172201, A108792.

Flatten[{{0, 0, 0, 0, 0, 0, 248, 7320, 82758},FullSimplify[Table[1/120*n^10-5/18*n^9+253/72*n^8-689/45*n^7-34217/360*n^6+28391/18*n^5-6828569/810*n^4+29655659/1620*n^3+14328773/1296*n^2-779503661/6480*n+9261910451/64800 +(1/8*n^5-143/48*n^4+79/3*n^3-4711/48*n^2+5171/48*n+2549/32)*(-1)^n +1/2*(29*n-35)*Cos[Pi*n/2] +(2*n+15)*Sin[Pi*n/2] +1/81*(96*n^3-1328*n^2+4744*n-2248)*Cos[4*Pi*n/3] -1/243*(120*n^2-1496*n+5224)*Sqrt[3]*Sin[4*Pi*n/3] +8/25*((5-Sqrt[5])*n+2*Sqrt[5]-8)*Cos[4*Pi*n/5] +8/25*((5+Sqrt[5])*n-2*Sqrt[5]-8)*Cos[8*Pi*n/5] +8/25*Sqrt[50-22*Sqrt[5]]*Sin[4*Pi*n/5] -8/25*Sqrt[50+22*Sqrt[5]]*Sin[8*Pi*n/5], {n, 10, 20}]]}]

a(n) = 1/120*n^10-5/18*n^9+253/72*n^8-689/45*n^7-34217/360*n^6+28391/18*n^5-6828569/810*n^4+29655659/1620*n^3+14328773/1296*n^2-779503661/6480*n+9261910451/64800 +(1/8*n^5-143/48*n^4+79/3*n^3-4711/48*n^2+5171/48*n+2549/32)*(-1)^n +1/2*(29*n-35)*cos(Pi*n/2) +(2*n+15)*sin(Pi*n/2) +1/81*(96*n^3-1328*n^2+4744*n-2248)*cos(4*Pi*n/3) -1/243*(120*n^2-1496*n+5224)*sqrt(3)*sin(4*Pi*n/3) +8/25*((5-sqrt(5))*n+2*sqrt(5)-8)*cos(4*Pi*n/5) +8/25*((5+sqrt(5))*n-2*sqrt(5)-8)*cos(8*Pi*n/5) +8/25*sqrt(50-22*sqrt(5))*sin(4*Pi*n/5) -8/25*sqrt(50+22*sqrt(5))*sin(8*Pi*n/5), n>=10.
a(n) = n^10/120 - 5*n^9/18 + 253*n^8/72 - 689*n^7/45 - 34307*n^6/360 + 57001*n^5/36 - 55000657*n^4/6480 + 60118543*n^3/3240 + 34387307*n^2/3240 - 155720509*n/1296 + 142960 + (n^5/2 - 143*n^4/12 + 316*n^3/3 - 4711*n^2/12 + 5123*n/12 + 2309/8)*floor[n/2] + (32*n^3/9 - 1328*n^2/27 + 4744*n/27 - 2248/27)*floor[n/3] + (16*n^3/9 - 724*n^2/27 + 1040*n/9 - 3736/27)*floor[(n+1)/3] + (33*n - 5)*floor[n/4] + (25*n - 65)*floor[(n+1)/4] + (32*n/5 - 48/5)*floor[n/5] + (24*n/5 - 64/5)*floor[(n+1)/5] + (16*n/5 - 56/5)*floor[(n+2)/5] + (8*n/5 - 32/5)*floor[(n+3)/5], n>=10.
G.f.: (2*x^7*(-124 - 3784*x - 44667*x^2 - 310723*x^3 - 1509124*x^4 - 5621180*x^5 - 16954312*x^6 - 42976662*x^7 - 93896850*x^8 - 180088868*x^9 - 307206501*x^10 - 470650261*x^11 - 652017897*x^12 - 820670989*x^13 - 941074901*x^14 - 984212615*x^15 - 938015444*x^16 - 812413066*x^17 - 635893628*x^18 - 445615046*x^19 - 275100707*x^20 - 145295581*x^21 - 61597137*x^22 - 17181649*x^23 + 704005*x^24 + 4589289*x^25 + 3324134*x^26 + 1424132*x^27 + 316332*x^28 - 58210*x^29 - 91844*x^30 - 47684*x^31 - 15863*x^32 - 3119*x^33 + 490*x^34 + 982*x^35 + 632*x^36 + 260*x^37 + 126*x^38 + 54*x^39))/((-1+x)^11*(1+x)^6*(1+x^2)^2*(1+x+x^2)^4*(1+x+x^2+x^3+x^4)^2).
Recurrence: a(n) = a(n-37) + a(n-36) - 3a(n-35) - 7a(n-34) - 3a(n-33) + 11a(n-32) + 21a(n-31) + 13a(n-30) - 13a(n-29) - 41a(n-28) - 44a(n-27) - 8a(n-26) + 49a(n-25) + 81a(n-24) + 57a(n-23) - 15a(n-22) - 88a(n-21) - 106a(n-20) - 48a(n-19) + 48a(n-18) + 106a(n-17) + 88a(n-16) + 15a(n-15) - 57a(n-14) - 81a(n-13) - 49a(n-12) + 8a(n-11) + 44a(n-10) + 41a(n-9) + 13a(n-8) - 13a(n-7) - 21a(n-6) - 11a(n-5) + 3a(n-4) + 7a(n-3) + 3a(n-2) - a(n-1), n>=47.

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A225553 Longest checkmate in king and amazon versus king endgame on an n X n chessboard.

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A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board.

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A173214 Number of ways to place 4 nonattacking amazons (superqueens) on an n X n board.

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A174642 Number of ways to place 4 nonattacking amazons (superqueens) on a 4 X n board.

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A178972 Number of ways to place 2 nonattacking amazons (superqueens) on an n X n toroidal board.

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A178967 Number of ways to place 5 nonattacking amazons (superqueens) on an n X n board.

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