A178967 Number of ways to place 5 nonattacking amazons (superqueens) on an n X n board.
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
Links
- V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
- Wikipedia, Amazon (chess)
Programs
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Mathematica
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}]]}]
Formula
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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