A190398 -id:A190398 - cp's OEIS frontend

A190400 Number of ways to place 5 nonattacking grasshoppers on a toroidal chessboard of size n x n.

0, 0, 0, 976, 18510, 201528, 1232448, 5637824, 20502396, 63720920, 174647286, 434439792, 997037470, 2141831160, 4348204020, 8412482304, 15605151496, 27903377016, 48291880442, 81188237680, 132977239290, 212739639640

Offset: 1

Vaclav Kotesovec, May 10 2011

The Grasshopper moves on the same lines as a queen, but must jump over a hurdle to land on the square immediately beyond.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Cf. A190397, A190398, A190399, A173775.

CoefficientList[Series[2 x^3 (144 x^18 - 874 x^17 + 1356 x^16 + 2195 x^15 - 8778 x^14 + 4282 x^13 + 16170 x^12 - 23696 x^11 - 5686 x^10 + 36079 x^9 - 33008 x^8 - 33909 x^7 - 13310 x^6 - 61448 x^5 - 197358 x^4 - 109070 x^3 - 50114 x^2 - 6327 x- 488) / ((x - 1)^11 (x + 1)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 03 2013 *)

Explicit formula: a(n) = 1/120*n^2*(n^8 -10*n^6 -240*n^5 +995*n^4 +640*n^3 +1870*n^2 -41680*n +69624) + 2*n^2*(n-3)*(2n+1)*(-1)^n, n>6.

G.f.: 2*x^4*(144*x^18 -874*x^17 +1356*x^16 +2195*x^15 -8778*x^14 +4282*x^13 +16170*x^12 -23696*x^11 -5686*x^10 +36079*x^9 -33008*x^8 -33909*x^7 -13310*x^6 -61448*x^5 -197358*x^4 -109070*x^3 -50114*x^2 -6327*x -488)/((x-1)^11*(x+1)^5).

A190399 Number of ways to place 4 nonattacking grasshoppers on a toroidal chessboard of size n x n.

Original entry on oeis.org

0, 1, 54, 1068, 8550, 45873, 177968, 562032, 1519560, 3662625, 8057390, 16477020, 31712850, 58018793, 101639700, 171525568, 280160068, 444636297, 687881890, 1040201500, 1541008350, 2240952065, 3204279960, 4511682288

Offset: 1

Vaclav Kotesovec, May 10 2011

The Grasshopper moves on the same lines as a queen, but must jump over a hurdle to land on the square immediately beyond.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Cf. A190396, A190398, A172519.

CoefficientList[Series[- x (80 x^14 - 444 x^13 + 768 x^12 + 108 x^11 - 1824 x^10 + 1600 x^9 + 1025 x^8 - 1200 x^7 + 708 x^6 + 1772 x^5 + 7254 x^4 + 2788 x^3 + 756 x^2 + 48 x + 1) / ((x - 1)^9 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 03 2013 *)

a(n) = 1/24*n^2*(n^6 -6*n^4 -96*n^3 +347*n^2 +96*n -726 +96*(-1)^n), n>4.

G.f.: -x^2*(80*x^14 -444*x^13 +768*x^12 +108*x^11 -1824*x^10 +1600*x^9 +1025*x^8 -1200*x^7 +708*x^6 +1772*x^5 +7254*x^4 +2788*x^3 +756*x^2 +48*x +1)/((x-1)^9*(x+1)^3).

A190401 Number of ways to place 6 nonattacking grasshoppers on a toroidal chessboard of size n x n.

Original entry on oeis.org

0, 0, 0, 384, 19100, 557808, 5841780, 41338400, 209264796, 859752800, 2982181004, 9131392296, 25196132260, 63968987264, 151223202900, 336724832384, 711538908572, 1437022315440, 2787781494732, 5219454908200, 9464698212228

Offset: 1

Vaclav Kotesovec, May 10 2011

The Grasshopper moves on the same lines as a queen, but must jump over a hurdle to land on the square immediately beyond.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Cf.: A190398, A190399, A190400.

CoefficientList[Series[- 4 x^3 (128 x^24 - 768 x^23 - 8 x^22 + 9258 x^21 - 17442 x^20 - 25593 x^19 + 103542 x^18 - 19695 x^17 - 252858 x^16 + 225766 x^15 + 297416 x^14 - 465166 x^13 - 63474 x^12 + 488076 x^11 + 515008 x^10 + 582376 x^9 + 2358586 x^8 + 2976026 x^7 + 6280504 x^6 + 4731396 x^5 + 2785972 x^4 + 664045 x^3 + 111570 x^2 + 4199 x + 96) / ((x - 1)^13 (x + 1)^7), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 03 2013 *)

Explicit formula: a(n) = 1/720*n^2*(n^10 -15n^8 -480n^7 +2245n^6 + 2400n^5 +27255n^4 -342480n^3 +639934n^2 +471600n -865080) + 1/4*n^2*(-1)^n*(8n^4 -40n^3 +17n^2 -272n +1608), n>8.

G.f.: -4x^4*(128x^24 -768x^23 -8x^22 +9258x^21 -17442x^20 -25593x^19 +103542x^18 -19695x^17 -252858x^16 +225766x^15 +297416x^14 -465166x^13 -63474x^12 +488076x^11 +515008x^10 +582376x^9 +2358586x^8 +2976026x^7 +6280504x^6 +4731396x^5 +2785972x^4 +664045x^3 +111570x^2 +4199x +96)/((x-1)^13*(x+1)^7).

cp's OEIS Frontend

A190400 Number of ways to place 5 nonattacking grasshoppers on a toroidal chessboard of size n x n.

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A190399 Number of ways to place 4 nonattacking grasshoppers on a toroidal chessboard of size n x n.

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A190401 Number of ways to place 6 nonattacking grasshoppers on a toroidal chessboard of size n x n.

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