A243716 -id:A243716 - cp's OEIS frontend

A243717 Number of inequivalent (mod D_4) ways to place 2 nonattacking knights on an n X n board.

2, 7, 18, 43, 83, 156, 257, 418, 624, 925, 1292, 1797, 2393, 3178, 4083, 5236, 6542, 8163, 9974, 12175, 14607, 17512, 20693, 24438, 28508, 33241, 38352, 44233, 50549, 57750, 65447, 74152, 83418, 93823, 104858, 117171, 130187, 144628, 159849, 176650, 194312

Offset: 2

Heinrich Ludwig, Jun 10 2014

Rotations or reflections of a placement are considered as the same. If they are distinguished, numbers are A172132.

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A243716, A172132, A243718, A243719, A243720.

[ (-33+(-1)^n+4*(9+(-1)^n)*n-2*(1+(-1)^n)*n^2+2*n^4)/32: n in [2..50]]; // Wesley Ivan Hurt, Jun 11 2014

A243717:=n->(-33+(-1)^n+4*(9+(-1)^n)*n-2*(1+(-1)^n)*n^2+2*n^4)/32; seq(A243717(n), n=2..50); # Wesley Ivan Hurt, Jun 11 2014

Table[(-33 + (-1)^n + 4*(9 + (-1)^n)*n - 2*(1 + (-1)^n)*n^2 + 2*n^4)/
32, {n, 2, 50}] (* Wesley Ivan Hurt, Jun 11 2014 *)

Vec(x^2*(x^6-3*x^4-5*x^3-3*x-2)/((x-1)^5*(x+1)^3) + O(x^100)) \\ Colin Barker, Jun 10 2014

a(n) = (n^4 - 2*n^2 + 20*n - 16 + IF(MOD(n, 2) = 1)*(2*n^2 - 4*n - 1))/16.
a(n) = (-33+(-1)^n+4*(9+(-1)^n)*n-2*(1+(-1)^n)*n^2+2*n^4)/32. - Colin Barker, Jun 10 2014
G.f.: x^2*(x^6-3*x^4-5*x^3-3*x-2) / ((x-1)^5*(x+1)^3). - Colin Barker, Jun 10 2014

A243718 Number of inequivalent (mod D_8) ways to place 3 nonattacking knights on an n X n board.

Original entry on oeis.org

1, 9, 40, 195, 618, 1751, 4075, 8794, 17015, 31268, 53666, 88781, 140200, 215405, 320013, 465436, 659965, 920114, 1257580, 1695303, 2249206, 2950131, 3819135, 4896590, 6209683, 7810096, 9732230, 12041009, 14779220, 18027113, 21837121, 26307056, 31500345

Offset: 2

Heinrich Ludwig, Jun 19 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

Cf. A243716, A172134, A243717, A243719, A243720.

Drop[CoefficientList[Series[-25 - 8*x + 3*x^3 + (25 - 67*x - 48*x^2 + 270*x^3 - 41*x^4 - 318*x^5 + 291*x^6 + 354*x^7 - 188*x^8 - 87*x^9 + 49*x^10) / ((1-x)^7*(1+x)^4), {x, 0, 20}], x],2] (* Vaclav Kotesovec, Jun 19 2014 *)

a(n) = (n^6 - 27*n^4 + 80*n^3 + 158*n^2 - 1028*n + 1200 + (1 - (-1)^n)/2*(8*n^3 - 9*n^2 - 44*n + 45))/48 for n >= 4.

G.f.: -25 - 8*x + 3*x^3 + (25 - 67*x - 48*x^2 + 270*x^3 - 41*x^4 - 318*x^5 + 291*x^6 + 354*x^7 - 188*x^8 - 87*x^9 + 49*x^10) / ((1-x)^7*(1+x)^4). - Vaclav Kotesovec, Jun 19 2014

A243719 Number of inequivalent (mod D_8) ways to place 4 nonattacking knights on an n X n board.

Original entry on oeis.org

1, 6, 66, 609, 3375, 14181, 47485, 136085, 342739, 784059, 1653033, 3267471, 6107271, 10901405, 18683285, 30934341, 49659915, 77611995, 118386689, 176753639, 258774303, 372270981, 526962861, 735113445, 1011678595, 1375177451, 1847843545, 2456771055, 3234056439

Offset: 2

Heinrich Ludwig, Jun 19 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).

Cf. A243716, A172135, A243717, A243718, A243720.

[1,6,66,609] cat [(n^8 - 54*n^6 + 144*n^5 + 1048*n^4 - 5280*n^3 - 2432*n^2 + 52800*n - 78912 + (1 - (-1)^n)/2*(14*n^4 - 48*n^3 - 158*n^2 + 768*n - 723))/192: n in [6..30]]; // Vincenzo Librandi, Jun 21 2014

Drop[CoefficientList[Series[411 + 171*x + 38*x^2 - 5*x^3 - 15*x^4 - 6*x^5 - (411 - 1473*x - 236*x^2 + 6588*x^3 - 5073*x^4 - 11179*x^5 + 13200*x^6 + 4572*x^7 - 19047*x^8 - 991*x^9 + 9564*x^10 - 1776*x^11 - 1955*x^12 + 675*x^13) / ((1-x)^9*(1+x)^5), {x, 0, 20}], x],2] (* Vaclav Kotesovec, Jun 19 2014 *)

a(n) = (n^8 - 54*n^6 + 144*n^5 + 1048*n^4 - 5280*n^3 - 2432*n^2 + 52800*n - 78912 + (1 - (-1)^n)/2*(14*n^4 - 48*n^3 - 158*n^2 + 768*n - 723))/192 for n >= 6.

G.f.: 411 + 171*x + 38*x^2 - 5*x^3 - 15*x^4 - 6*x^5 - (411 - 1473*x - 236*x^2 + 6588*x^3 - 5073*x^4 - 11179*x^5 + 13200*x^6 + 4572*x^7 - 19047*x^8 - 991*x^9 + 9564*x^10 - 1776*x^11 - 1955*x^12 + 675*x^13) / ((1-x)^9*(1+x)^5). - Vaclav Kotesovec, Jun 19 2014

A243720 Number of inequivalent (mod D_8) ways to place 5 nonattacking knights on an n X n board.

Original entry on oeis.org

2, 49, 1244, 12329, 81900, 398907, 1562362, 5153001, 14907120, 38753358, 92417760, 204977323, 427812496, 847346181, 1604300270

Offset: 3

Heinrich Ludwig, Jun 19 2014

Cf. A243716, A172136, A243717, A243718, A243719.

a(n) = (n^10 - 90*n^8 + 240*n^7 + 3235*n^6)/960 + O(n^5) for n >= 8.

A243281 Number of inequivalent (mod D_8) ways to place n nonattacking knights on an n X n board.

Original entry on oeis.org

1, 2, 9, 66, 1244, 32524, 1114549, 47513547, 2401002993, 140026612346

Offset: 1

Heinrich Ludwig, Jun 19 2014

Table of n, a(n) for n = 1..10

Cf. A201540, A243716.

cp's OEIS Frontend

A243717 Number of inequivalent (mod D_4) ways to place 2 nonattacking knights on an n X n board.

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A243718 Number of inequivalent (mod D_8) ways to place 3 nonattacking knights on an n X n board.

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A243719 Number of inequivalent (mod D_8) ways to place 4 nonattacking knights on an n X n board.

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A243720 Number of inequivalent (mod D_8) ways to place 5 nonattacking knights on an n X n board.

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A243281 Number of inequivalent (mod D_8) ways to place n nonattacking knights on an n X n board.

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