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A178975 Number of ways to place 5 nonattacking amazons (superqueens) on an n X n toroidal board. 2

%I #21 Sep 12 2015 11:00:23

%S 0,0,0,0,0,0,0,640,7290,123640,640574,3869280,12950132,47022360,

%T 123467040,340840960,759697190,1758672648,3494388306,7150739360,

%U 13041285516,24354594440,41566378136,72345297024,117101090250,192694385416,298703838186,469581881888,702148696580,1062719841960,1541332566284,2259300468736,3192255589842,4552716843720,6288527141890,8758324830240,11859789616944,16178716174856,21527161542900,28834708173440

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

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

%H Vincenzo Librandi, <a href="/A178975/b178975.txt">Table of n, a(n) for n = 1..1000</a>

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013

%F 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.

%F 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).

%t 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 *)

%Y Cf. A178967, A173775, A178972, A178973, A178974.

%K nonn,nice,easy

%O 1,8

%A _Vaclav Kotesovec_, Jan 02 2011

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