A243718 -id:A243718 - cp's OEIS frontend

A243716 Irregular triangle read by rows: T(n, k) = number of inequivalent (mod the dihedral group D_8 of order 8) ways to place k nonattacking knights on an n X n board.

1, 1, 2, 1, 1, 3, 7, 9, 6, 2, 3, 18, 40, 66, 49, 30, 8, 3, 6, 43, 195, 609, 1244, 1767, 1710, 1148, 510, 154, 31, 6, 1, 6, 83, 618, 3375, 12329, 32524, 61731, 86748, 90059, 70128, 40770, 18053, 6089, 1643, 344, 61, 7, 1, 10, 156, 1751, 14181, 81900, 348541

Offset: 1

Heinrich Ludwig, Jun 10 2014

The triangle is irregularly shaped: 1 <= k <= A030978(n). A030978(n) is the maximal number of knights that can be placed on an n X n board.
First row corresponds to n = 1.
Counting "inequivalent ways" means: Rotations or reflections of a placement of knights on the board are considered to be the same placement.

			The triangle begins:
  1;
  1,  2,   1,   1;
  3,  7,   9,   6,    2;
  3, 18,  40,  66,   49,   30,    8,    3;
  6, 43, 195, 609, 1244, 1767, 1710, 1148, 510, 154, 31, 6, 1;
  ...

Heinrich Ludwig, Table of n, a(n) for n = 1..116

Cf. A030978, A008805 (column 1), A243717 (column 2), A243718 (column 3), A243719 (column 4), A243720 (column 5).

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

Original entry on oeis.org

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

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.

cp's OEIS Frontend

A243716 Irregular triangle read by rows: T(n, k) = number of inequivalent (mod the dihedral group D_8 of order 8) ways to place k nonattacking knights on an n X n board.

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A243717 Number of inequivalent (mod D_4) ways to place 2 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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