cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A190397 Number of ways to place 5 nonattacking grasshoppers on a chessboard of size n x n.

Original entry on oeis.org

0, 0, 28, 1668, 29092, 252584, 1441634, 6222996, 22004086, 66972760, 181332416, 446905476, 1019470032, 2179712872, 4410518630, 8510498516, 15756224370, 28128603736, 48622240660, 81660504068, 133643402268, 213660267432
Offset: 1

Views

Author

Vaclav Kotesovec, May 10 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A190395, A190396, A108792.

Programs

  • Mathematica
    CoefficientList[Series[2 x^2 (8 x^12 - 60 x^11 + 75 x^10 + 24 x^9 + 441 x^8 - 1948 x^7 - 893 x^6 + 4122 x^5 - 8491 x^4 - 15988 x^3 - 6822 x^2 - 694 x - 14) / ((x - 1)^11 (x+1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

Formula

a(n) = 1/120*(n^10 -10*n^8 -200*n^7 +1175*n^6 -1136*n^5 -740*n^4 -30520*n^3 +159624*n^2 -289024*n +179175 -135*(-1)^n), n>3.
G.f.: 2x^3*(8*x^12 -60*x^11 +75*x^10 +24*x^9 +441*x^8 -1948*x^7 -893*x^6 +4122*x^5 -8491*x^4 -15988*x^3 -6822*x^2 -694*x -14)/((x-1)^11*(x+1)).

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

Views

Author

Vaclav Kotesovec, May 10 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A190396, A190398, A172519.

Programs

  • Mathematica
    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 *)

Formula

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).

A190579 Number of ways to place 6 nonattacking grasshoppers on an n x n chessboard.

Original entry on oeis.org

0, 0, 2, 998, 51618, 873852, 8039322, 50272978, 240764814, 947860554, 3210392210, 9649651136, 26316155354, 66191981440, 155482089002, 344411086374, 725043524246, 1459722296638, 2825136685698, 5278863810724, 9557560367842
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A190395, A190396, A190397, A176186, A190401.

Programs

  • Mathematica
    CoefficientList[Series[2 x^2 (8 x^18 - 59 x^17 + 110 x^16 + 71 x^15 + 473 x^14 - 3017 x^13 - 5401 x^12 + 23838 x^11 - 2727 x^10 - 119474 x^9 - 45545 x^8 - 20157 x^7 - 571677 x^6 - 1006961 x^5 - 689547 x^4 - 199704 x^3 - 20861 x^2 - 489 x - 1) / ((x - 1)^13 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 03 2013 *)

Formula

a(n) = n^12/720 -n^10/48 -5n^9/9 +509n^8/144 -187n^7/90 +701n^6/48 -14467n^5/36 +666917n^4/360 -471121n^3/180 -59875n^2/24 +57101n/6 -11339/2 -(9n^2/8-n-7/2)*(-1)^n, n>5.
G.f.: 2x^3*(8x^18 -59x^17 +110x^16 +71x^15 +473x^14 -3017x^13 -5401x^12 +23838x^11 -2727x^10 -119474x^9 -45545x^8 -20157x^7 -571677x^6 -1006961x^5 -689547x^4 -199704x^3 -20861x^2 -489x -1)/((x-1)^13*(x+1)^3).
Showing 1-3 of 3 results.

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