A201244 -id:A201244 - cp's OEIS frontend

A201243 Number of ways to place 2 non-attacking ferses on an n X n board.

0, 4, 28, 102, 268, 580, 1104, 1918, 3112, 4788, 7060, 10054, 13908, 18772, 24808, 32190, 41104, 51748, 64332, 79078, 96220, 116004, 138688, 164542, 193848, 226900, 264004, 305478, 351652, 402868, 459480, 521854, 590368, 665412, 747388, 836710, 933804, 1039108

Offset: 1

Vaclav Kotesovec, Nov 28 2011

Fers is a leaper [1,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, p.415
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A172123, A201244, A201245, A201246, A201247, A201248.

I:=[0, 4, 28, 102, 268]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[(n-1)*(n^3+n^2-4*n+4)/2: n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

Table[(n - 1) (n^3 + n^2 - 4 n + 4) / 2, {n, 100}] (* Vincenzo Librandi, Apr 30 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,4,28,102,268},40] (* Harvey P. Dale, Dec 31 2014 *)

a(n) = 1/2*(n-1)*(n^3 + n^2 - 4n + 4) by C. Poisson, 1990.
G.f.: 2x^2*(x+1)*(x^2-2x-2)/(x-1)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Vincenzo Librandi, Apr 30 2013

A201245 Number of ways to place 4 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 29, 661, 6285, 35378, 143787, 468529, 1301351, 3202970, 7170593, 14872997, 28969129, 53527866, 94568255, 160741233, 264175507, 421511954, 655152581, 994751765, 1478979173, 2157585442, 3093803379, 4367119121, 6076449375, 8343762538, 11318183177

Offset: 1

Vaclav Kotesovec, Nov 28 2011

Fers is a leaper [1,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces, 6ed, p.415
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A172127, A201243, A201244, A201246, A201247, A201248.

CoefficientList[Series[- x^2 (2 x^8 - 55 x^7 + 230 x^6 - 254 x^5 - 225 x^4 + 173 x^3 + 1380 x^2 + 400 x + 29)/(x-1)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 30 2013 *)

a(n) = (n^8 - 30n^6 + 48n^5 + 299n^4 - 912n^3 - 462n^2 + 4368n - 4200)/24, n>=3.

G.f.: -x^3*(2*x^8 - 55*x^7 + 230*x^6 - 254*x^5 - 225*x^4 + 173*x^3 + 1380*x^2 + 400*x + 29)/(x-1)^9.

A201246 Number of ways to place 5 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 12, 780, 16286, 159452, 992412, 4567836, 16959488, 53617596, 149618794, 377841356, 879314442, 1911495356, 3922051616, 7657895196, 14321764860, 25791609308, 44921419134, 75946019596, 125016699158, 200899440924, 315872975684, 486869916572, 736910896536

Offset: 1

Vaclav Kotesovec, Nov 28 2011

Fers is a leaper [1,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces, 6ed, p.415
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A172129, A201243, A201244, A201245, A201247, A201248.

CoefficientList[Series[2 x^2 (11 x^11 - 135 x^10 + 549 x^9 - 993 x^8 + 1172 x^7 - 2968 x^6 + 7085 x^5 - 4715x^4 - 10613 x^3 - 4183 x^2- 324 x - 6)/(x-1)^11, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 30 2013 *)

a(n) = n^10/120 - 5n^8/12 + 2n^7/3 + 191n^6/24 - 24n^5 - 661n^4/12 + 880n^3/3 - 937n^2/15 - 1176n + 1436, n>=4.

G.f.: 2x^3*(11x^11 - 135x^10 + 549x^9 - 993x^8 + 1172x^7 - 2968x^6 + 7085x^5 - 4715x^4 - 10613x^3 - 4183x^2 - 324x - 6)/(x-1)^11.

A201247 Number of ways to place 6 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 2, 552, 29412, 527654, 5196928, 34528698, 173951172, 714042302, 2503447216, 7744201834, 21635290132, 55540293510, 132752090192, 298491879178, 636559136340, 1296099575166, 2533344878048, 4774975629082, 8712052571140, 15436347060646, 26634487077600

Offset: 1

Vaclav Kotesovec, Nov 28 2011

Fers is a leaper [1,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces, 6ed, p.415
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A176886, A201243, A201244, A201245, A201246, A201248.

CoefficientList[Series[- 2 x^2 (41 x^14 - 502 x^13 + 2506 x^12 - 7605 x^11 + 18870 x^10 - 41305 x^9 + 60117 x^8 - 21366 x^7 - 73987 x^6 + 52960 x^5 + 237560 x^4 + 93891 x^3 + 11196 x^2 + 263 x + 1)/(x-1)^13, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 30 2013 *)

a(n) = n^12/720 - 5n^10/48 + n^9/6 + 461n^8/144 - 29n^7/3 - 2147n^6/48 + 1289n^5/6 + 65807n^4/360 - 6356n^3/3 + 9185n^2/6 + 22834n/3 - 11478, n>=5.

G.f.: -2x^3*(41x^14 - 502x^13 + 2506x^12 - 7605x^11 + 18870x^10 - 41305x^9 + 60117x^8 - 21366x^7 - 73987x^6 + 52960x^5 + 237560x^4 + 93891x^3 + 11196x^2 + 263x + 1)/(x-1)^13.

A201248 Number of ways to place 7 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 0, 216, 38070, 1314600, 21191208, 207830308, 1442794332, 7775083960, 34530764200, 131660992164, 443702617356, 1350258600008, 3771242866680, 9789675562020, 23856321869260, 55015308882264, 120855465245464, 254284702668580, 514791197224860, 1006655249550696

Offset: 1

Vaclav Kotesovec, Nov 28 2011

nonn

Fers is a leaper [1,1].

V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).

Cf. A187239, A201243, A201244, A201245, A201246, A201247.

a(n) = n^14/5040 - n^12/48 + n^11/30 + 673n^10/720 - 17n^9/6 - 1019n^8/48 + 197n^7/2 + 9772n^6/45 - 3443n^5/2 + 47n^4/4 + 74259n^3/5 - 1816352n^2/105 - 49376n + 90660, n>=6.

G.f.: 2*x^4*(125*x^16 - 1785*x^15 + 11715*x^14 - 50121*x^13 + 158605*x^12 - 367485*x^11 + 570175*x^10 - 533381*x^9 + 460395*x^8 - 1262515*x^7 + 2731225*x^6 - 1795227*x^5 - 5484089*x^4 - 2685639*x^3 - 383115*x^2 - 17415*x - 108)/(x-1)^15.

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)

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A201243 Number of ways to place 2 non-attacking ferses on an n X n board.

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A201245 Number of ways to place 4 non-attacking ferses on an n X n board.

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A201246 Number of ways to place 5 non-attacking ferses on an n X n board.

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A201247 Number of ways to place 6 non-attacking ferses on an n X n board.

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A201248 Number of ways to place 7 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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