A278683 -id:A278683 - cp's OEIS frontend

A278688 Triangle read by rows T(n, k) = number of non-equivalent ways to place k non-attacking ferses on an n X n board.

1, 1, 1, 1, 1, 1, 3, 6, 7, 6, 2, 1, 1, 3, 17, 45, 92, 99, 76, 27, 7, 1, 6, 43, 225, 832, 2102, 3773, 4860, 4643, 3356, 1868, 795, 248, 56, 8, 1, 1, 6, 84, 709, 4500, 19987, 66201, 164423, 314224, 465230, 540247, 492206, 352300, 195717, 83247, 26083, 5754, 780, 55

Offset: 1

Heinrich Ludwig, Nov 27 2016

The triangle T(n, k) is irregularly shaped: 0 <= k <= A093005(n), which means that A093005(n) is the maximal number of non-attacking ferses that can be placed on an n X n board. First row corresponds to n = 1. First column corresponds to k = 0.
Two placements that differ by rotation or reflection are counted only once.
A fers is a fairy chess piece attacking one step ne-nw-sw-se.

			Triangle begins:
1, 1;
1, 1,  1;
1, 3,  6,  7,  6,  2,  1;
1, 3, 17, 45, 92, 99, 76, 27, 7;

Heinrich Ludwig, Table of n, a(n) for n = 1..130
Wikipedia, Fairy chess piece

Cf. A008805, A232567, A278682, A278683, A278684, A278685, A278686, (columns 2 through 8 of this sequence, respectively), A278687, A093005 (row length - 1).

A278682 Number of non-equivalent ways to place 3 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 7, 45, 225, 709, 1974, 4524, 9614, 18382, 33425, 56895, 93447, 146715, 224280, 331814, 480844, 679724, 945099, 1288737, 1733725, 2296065, 3006762, 3886960, 4977210, 6304794, 7921589, 9862099, 12191459, 14952567, 18225900, 22064010, 26564952, 31792280

Offset: 1

Heinrich Ludwig, Nov 26 2016

A fers is a leaper [1, 1].

Rotations and reflections of placements are not counted. If they are to be counted, see A201244.

			There are 7 ways to place 3 non-attacking ferses "X" on a 3 X 3 board, rotations and reflections being ignored
   XXX   XX.   X.X   ...   X..   X..   X..
   ...   ...   ...   XXX   X.X   ...   ...
   ...   ..X   .X.   ...   ...   XX.   X.X

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

Cf. A201244, A232567 (2 ferses), A278683 (4 ferses), A278684 (5 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.

Table[Boole[n > 2] ((n^6 - 15 n^4 + 32 n^3 + 14 n^2 - 116 n + 96) + Boole[OddQ@ n] (8 n^3 - 9 n^2 - 20 n + 9))/48, {n, 34}] (* Michael De Vlieger, Nov 30 2016 *)

concat(vector(2), Vec(x^3*(7 + 24*x + 83*x^2 + 66*x^3 + 75*x^4 - 15*x^6 - 2*x^7 + 2*x^8) / ((1 - x)^7*(1 + x)^4) + O(x^40))) \\ Colin Barker, Dec 07 2016

a(n) = ((n^6 - 15*n^4 + 32*n^3 + 14*n^2 - 116*n + 96) + IF(MOD(n, 2) = 1, 8*n^3 - 9*n^2 - 20*n + 9))/48.
a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11).
From Colin Barker, Dec 07 2016: (Start)
a(n) = (n^6 - 15*n^4 + 32*n^3 + 14*n^2 - 116*n + 96)/48 for n even.
a(n) = (n^6 - 15*n^4 + 40*n^3 + 5*n^2 - 136*n + 105)/48 for n odd.
G.f.: x^3*(7 + 24*x + 83*x^2 + 66*x^3 + 75*x^4 - 15*x^6 - 2*x^7 + 2*x^8) / ((1 - x)^7*(1 + x)^4).
(End)

A278684 Number of non-equivalent ways to place 5 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 2, 99, 2102, 19987, 124676, 571418, 2122841, 6704061, 18711691, 47235845, 109938296, 238950999, 490309398, 957267228, 1790325363, 3224010105, 5615368229, 9493358359, 15627413290, 25112609019, 39484650296, 60859027054, 92114682749, 137111560949, 200972392655

Offset: 1

Heinrich Ludwig, Nov 26 2016

A fers is a leaper [1, 1].

Rotations and reflections of placements are not counted. If they are to be counted, see A201246.

			There are 2 ways to place 5 non-attacking ferses on a 3 X 3 board, rotations and reflections being ignored:
   XXX   XXX
   ...   ...
   X.X   XX.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Wikipedia, Fairy chess piece
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Cf. A201246, A232567 (2 ferses), A278682 (3 ferses), A278683 (4 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.

[0, 0, 2] cat [(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960 : n in [4..30]]; // Wesley Ivan Hurt, Nov 27 2016

A278684:=n->(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960: 0, 0, 2, seq(A278684(n), n=4..30); # Wesley Ivan Hurt, Nov 27 2016

Join[{0, 0, 2}, Table[(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960, {n, 4, 30}]] (* Wesley Ivan Hurt, Nov 27 2016 *)

concat(vector(2), Vec(x^3*(2 +89*x +1615*x^2 +9913*x^3 +35049*x^4 +66034*x^5 +78731*x^6 +45748*x^7 +9902*x^8 -5540*x^9 -1343*x^10 +1685*x^11 +409*x^12 -334*x^13 -83*x^14 +38*x^15 +6*x^16 -x^17) / ((1 -x)^11*(1 +x)^6) + O(x^40))) \\ Colin Barker, Dec 10 2016

a(n) = (n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960 for n >= 4.
a(n) = 5*a(n-1) - 4*a(n-2) - 20*a(n-3) + 40*a(n-4) + 16*a(n-5) - 100*a(n-6) + 44*a(n-7) + 110*a(n-8) - 110*a(n-9) - 44*a(n-10) + 100*a(n-11) - 16*a(n-12) - 40*a(n-13) + 20*a(n-14) + 4*a(n-15) - 5*a(n-16) + a(n-17) for n >= 21.
G.f.: x^3*(2 +89*x +1615*x^2 +9913*x^3 +35049*x^4 +66034*x^5 +78731*x^6 +45748*x^7 +9902*x^8 -5540*x^9 -1343*x^10 +1685*x^11 +409*x^12 -334*x^13 -83*x^14 +38*x^15 +6*x^16 -x^17) / ((1 -x)^11*(1 +x)^6). - Colin Barker, Dec 10 2016

A278685 Number of non-equivalent ways to place 6 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 1, 76, 3773, 66201, 651193, 4318451, 21754341, 89267490, 312974387, 968069337, 2704548145, 6942663519, 16594368633, 37311795887, 79570707969, 162013125016, 316669793867, 596873304925, 1089009784181, 1929545889877, 3329316638249, 5607471933963, 9238336533613

Offset: 1

Heinrich Ludwig, Nov 26 2016

A fers is a leaper [1, 1].

Rotations and reflections of placements are not counted. If they are to be counted, see A201247.

			There is 1 way to place 6 non-attacking ferses on a 3 X 3 board, rotations and reflections being ignored:
   XXX
   ...
   XXX

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Wikipedia, Fairy chess piece
Index entries for linear recurrences with constant coefficients, signature (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).

Cf. A201247, A232567 (2 ferses), A278682 (3 ferses), A278683 (4 ferses), A278684 (5 ferses), A278686 (7 ferses), A278687, A278688.

concat(vector(2), Vec(x^3*(1 +70*x +3325*x^2 +44193*x^3 +285774*x^4 +1018671*x^5 +2250048*x^6 +3090821*x^7 +2658486*x^8 +1198906*x^9 +139256*x^10 -84845*x^11 +22114*x^12 +28024*x^13 -6172*x^14 -5377*x^15 +485*x^16 +592*x^17 +199*x^18 -56*x^19 -44*x^20 +9*x^21) / ((1 -x)^13*(1 +x)^7) + O(x^30))) \\ Colin Barker, Dec 10 2016

a(n) = n^12 - 75*n^10 + 120*n^9 + 2305*n^8 - 6960*n^7 - 32008*n^6 + 152880*n^5 + 138204*n^4 - 1543560*n^3 + 1178528*n^2 + 5238720*n - 7977600 + IF(MOD(n, 2) = 1, 122*n^6 - 600*n^5 - 1645*n^4 + 14520*n^3 - 19447*n^2 - 30480*n + 81855)/5760 for n>=5.
a(n) = 6*a(n-1)-8*a(n-2)-22*a(n-3)+69*a(n-4)-8*a(n-5)-176*a(n-6)+168*a(n-7)+182*a(n-8)-364*a(n-9)+364*a(n-11)-182*a(n-12)-168*a(n-13)+176*a(n-14)+8*a(n-15)-69*a(n-16)+22*a(n-17)+8*a(n-18)-6*a(n-19)+*a(n-20) for n>=25.
G.f.: x^3*(1 +70*x +3325*x^2 +44193*x^3 +285774*x^4 +1018671*x^5 +2250048*x^6 +3090821*x^7 +2658486*x^8 +1198906*x^9 +139256*x^10 -84845*x^11 +22114*x^12 +28024*x^13 -6172*x^14 -5377*x^15 +485*x^16 +592*x^17 +199*x^18 -56*x^19 -44*x^20 +9*x^21) / ((1 -x)^13*(1 +x)^7). - Colin Barker, Dec 10 2016

A278686 Number of non-equivalent ways to place 7 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 0, 27, 4860, 164423, 2651890, 25981150, 180378380, 971905679, 4316504623, 16457726539, 55463445891, 168782705327, 471407278652, 1223710587908, 2982045310010

Offset: 1

Heinrich Ludwig, Dec 02 2016

A fers is a leaper [1, 1].

Rotations and reflections of placements are not counted. If they are to be counted, see A201248.

			There are 27 non-equivalent ways to place 7 non-attacking ferses (X) on a 4 X 4 board, rotations and reflections being ignored, e.g., these two:
   XXXX   X.XX
   ....   ....
   XXX.   X.X.
   ....   X.X.

Wikipedia, Fairy chess piece

Cf. A201248, A232567 (2 ferses), A278682 (3 ferses), A278683 (4 ferses), A278684 (5 ferses), A278685 (6 ferses), A278687, A278688.

cp's OEIS Frontend

A278688 Triangle read by rows T(n, k) = number of non-equivalent ways to place k non-attacking ferses on an n X n board.

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A278682 Number of non-equivalent ways to place 3 non-attacking ferses on an n X n board.

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A278684 Number of non-equivalent ways to place 5 non-attacking ferses on an n X n board.

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A278685 Number of non-equivalent ways to place 6 non-attacking ferses on an n X n board.

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A278686 Number of non-equivalent ways to place 7 non-attacking ferses on an n X n board.

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