A201247 -id:A201247 - 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

A201244 Number of ways to place 3 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 38, 340, 1630, 5552, 15210, 35828, 75530, 146240, 264702, 453620, 742918, 1171120, 1786850, 2650452, 3835730, 5431808, 7545110, 10301460, 13848302, 18357040, 24025498, 31080500, 39780570, 50418752, 63325550, 78871988, 97472790, 119589680, 145734802

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 (7,-21,35,-35,21,-7,1).

Cf. A172124, A201243, A201245, A201246, A201247, A201248.

I:=[0, 0, 38, 340, 1630, 5552, 15210, 35828]; [n le 8 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[0] cat [(n-2)*(n^5+2*n^4-11*n^3 +2*n^2+54*n-60)/6: n in [2..35]]; // Vincenzo Librandi, Apr 30 2013

CoefficientList[Series[- 2 x^2 (x^5 + 3 x^4 - 24 x^3 + 24 x^2 + 37 x + 19) / (x-1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 30 2013 *)

a(n) = (n-2)*(n^5 + 2n^4 - 11n^3 + 2n^2 + 54n - 60)/6, n>=2.
G.f.: -2x^3*(x^5 + 3x^4 - 24x^3 + 24x^2 + 37x + 19)/(x-1)^7.
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - 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.

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.

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

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

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A201244 Number of ways to place 3 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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A201248 Number of ways to place 7 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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