A178967 -id:A178967 - 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

A178975 Number of ways to place 5 nonattacking amazons (superqueens) on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 640, 7290, 123640, 640574, 3869280, 12950132, 47022360, 123467040, 340840960, 759697190, 1758672648, 3494388306, 7150739360, 13041285516, 24354594440, 41566378136, 72345297024, 117101090250, 192694385416, 298703838186, 469581881888, 702148696580, 1062719841960, 1541332566284, 2259300468736, 3192255589842, 4552716843720, 6288527141890, 8758324830240, 11859789616944, 16178716174856, 21527161542900, 28834708173440

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. A178967, A173775, A178972, A178973, A178974.

CoefficientList[Series[- 2 x^7 * (1176 x^64 + 5556 x^63 + 15132 x^62 + 28428 x^61 + 39340 x^60 + 30066 x^59 - 16046 x^58 - 97562 x^57 - 191158 x^56 - 227584 x^55 - 150082 x^54 + 56017 x^53 + 289119 x^52 + 339896 x^51 + 45336 x^50 - 611255 x^49 - 1380704 x^48 - 2278261 x^47 - 3764650 x^46 - 7542849 x^45 - 7704482 x^44 + 18495516 x^43 + 165924351 x^42 + 637466559 x^41 + 1903273538 x^40 + 4724140916 x^39 + 10422040024 x^38 + 20690172375 x^37 + 37875420877 x^36 + 64238796480 x^35 + 102190978070 x^34 + 152823563437 x^33 + 216401077492 x^32 + 290462738417 x^31 + 371272897408 x^30 + 452086367452 x^29 + 526060962825 x^28 + 584865148004 x^27 + 622627590675 x^26 + 634259897550 x^25 + 619201117902 x^24 + 578669435625 x^23 + 518210895306 x^22 + 443951015905 x^21 + 364069798686 x^20 + 285127462600 x^19 + 213313173667 x^18 + 151952471981 x^17 + 103062047860 x^16 + 66251579160 x^15 + 40354587182 x^14 + 23135311545 x^13 + 12479773177 x^12 + 6269223018 x^11 + 2933204824 x^10 + 1256492269 x^9 + 493760966 x^8 + 172473531 x^7 + 54013568 x^6 + 14176791 x^5 + 3222186 x^4 + 525572 x^3 + 74355 x^2 + 4605 x + 320) / ((x - 1)^11 (x + 1)^9 (x^2 + 1)^5 (x^2 - x + 1)^3 (x^2 + x + 1)^5 (x^4 + x^3 + x^2 + x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)

a(n) = n^2/5*(1/24*n^8 -5/3*n^7 +635/24*n^6 -1145/6*n^5 +4139/16*n^4 +10985/2*n^3 -916385/24*n^2 +273775/3*n -128930/3 +(5/24*n^6 -15/2*n^5 +5315/48*n^4 -1675/2*n^3 +25855/8*n^2 -4935*n -2290/3)*(-1)^n +(45/2*n^2 -390*n +1630)*cos(n*Pi/2) +400/3*cos(n*Pi/3) +(40/3*n^2 -640/3*n+2480/3)*cos(2*n*Pi/3) +16*cos(4*n*Pi/5) +16*cos(8*n*Pi/5)), n>=15.

G.f.: -2*x^8*(1176*x^64 + 5556*x^63 + 15132*x^62 + 28428*x^61 + 39340*x^60 + 30066*x^59 - 16046*x^58 - 97562*x^57 - 191158*x^56 - 227584*x^55 - 150082*x^54 + 56017*x^53 + 289119*x^52 + 339896*x^51 + 45336*x^50 - 611255*x^49 - 1380704*x^48 - 2278261*x^47 - 3764650*x^46 - 7542849*x^45 - 7704482*x^44 + 18495516*x^43 + 165924351*x^42 + 637466559*x^41 + 1903273538*x^40 + 4724140916*x^39 + 10422040024*x^38 + 20690172375*x^37 + 37875420877*x^36 + 64238796480*x^35 + 102190978070*x^34 + 152823563437*x^33 + 216401077492*x^32 + 290462738417*x^31 + 371272897408*x^30 + 452086367452*x^29 + 526060962825*x^28 + 584865148004*x^27 + 622627590675*x^26 + 634259897550*x^25 + 619201117902*x^24 + 578669435625*x^23 + 518210895306*x^22 + 443951015905*x^21 + 364069798686*x^20 + 285127462600*x^19 + 213313173667*x^18 + 151952471981*x^17 + 103062047860*x^16 + 66251579160*x^15 + 40354587182*x^14 + 23135311545*x^13 + 12479773177*x^12 + 6269223018*x^11 + 2933204824*x^10 + 1256492269*x^9 + 493760966*x^8 + 172473531*x^7 + 54013568*x^6 + 14176791*x^5 + 3222186*x^4 + 525572*x^3 + 74355*x^2 + 4605*x + 320)/((x-1)^11*(x+1)^9*(x^2+1)^5*(x^2-x+1)^3*(x^2+x+1)^5*(x^4+x^3+x^2+x+1)^3).

cp's OEIS Frontend

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

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