A278688 -id:A278688 - cp's OEIS frontend

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

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)

A278683 Number of non-equivalent ways to place 4 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 6, 92, 832, 4500, 18229, 58881, 163509, 401259, 898420, 1861146, 3625546, 6694982, 11829267, 20099815, 33036079, 52700901, 81916834, 124362664, 184907220, 269726216, 386776561, 545930397, 759628777, 1043027055, 1414873104, 1897655046, 2518755934, 3310591194

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 A201245.

			There are 6 ways to place 4 non-attacking ferses on a 3 X 3 board rotations and reflections being ignored:
   XXX   XXX   X.X   X.X   XX.   XX.
   ...   ...   ...   ...   ...   ...
   ..X   .X.   X.X   XX.   XX.   .XX

Heinrich Ludwig, Table of n, a(n) for n = 1..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. A201245, A232567 (2 ferses), A278682 (3 ferses), A278684 (5 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.

Table[Boole[n > 2] (n^8 - 30 n^6 + 48 n^5 + 328 n^4 - 1056 n^3 - 200 n^2 + 4176 n - 4032 + Boole[OddQ@ n] (14 n^4 - 48 n^3 - 38 n^2 + 336 n - 459))/192, {n, 30}] (* Michael De Vlieger, Nov 30 2016 *)

concat(vector(2), Vec(x^3*(6 +68*x +470*x^2 +1360*x^3 +2419*x^4 +1909*x^5 +836*x^6 -232*x^7 -192*x^8 +30*x^9 +54*x^10 -9*x^12 +x^13) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ Colin Barker, Dec 10 2016

a(n) = (n^8 - 30*n^6 + 48*n^5 + 328*n^4 - 1056*n^3 - 200*n^2 + 4176*n - 4032 + IF(MOD(n, 2) = 1, 14*n^4 - 48*n^3 - 38*n^2 + 336*n - 459))/192 for n>=3.
a(n) = 4*a(n-1) - a(n-2) - 16*a(n-3) + 19*a(n-4) + 20*a(n-5) - 45*a(n-6) + 45*a(n-8) - 20*a(n-9) - 19*a(n-10) + 16*a(n-11) + a(n-12) - 4*a(n-13) + a(n-14) for n>=17.
G.f.: x^3*(6 +68*x +470*x^2 +1360*x^3 +2419*x^4 +1909*x^5 +836*x^6 -232*x^7 -192*x^8 +30*x^9 +54*x^10 -9*x^12 +x^13) / ((1 -x)^9*(1 +x)^5). - Colin Barker, Dec 10 2016

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

A278687 Number of non-equivalent ways to place n non-attacking ferses on an n X n board.

Original entry on oeis.org

1, 1, 7, 92, 2102, 66201, 2651890, 126928146, 7060794663, 447024321962

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 A201861.

Wikipedia, Fairy chess piece

Cf. A201861, A278688.

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.

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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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A278683 Number of non-equivalent ways to place 4 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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A278687 Number of non-equivalent ways to place n 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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