A172213 -id:A172213 - cp's OEIS frontend

A172214 Number of ways to place 5 nonattacking knights on a 5 X n board.

1, 28, 259, 1968, 9386, 30842, 82738, 192336, 400277, 763984, 1360797, 2291056, 3681226, 5687022, 8496534, 12333352, 17459691, 24179516, 32841667, 43842984, 57631432, 74709226, 95635956, 121031712, 151580209

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A172136, A061991, A172212, A172213.

CoefficientList[Series[(42 x^12 - 52 x^11 - 268 x^10 + 884 x^9 - 268 x^8 - 1188 x^7 + 2834 x^6 - 720 x^5 + 918 x^4 + 814 x^3 + 106 x^2 + 22 x + 1) / (x - 1)^6, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (625n^5-8250n^4+57235n^3-242778n^2+608440n-705984)/24, n>=8.

G.f.: x * (42*x^12 -52*x^11 -268*x^10 +884*x^9 -268*x^8 -1188*x^7 +2834*x^6 -720*x^5 +918*x^4 +814*x^3 +106*x^2 +22*x +1) / (x-1)^6. - Vaclav Kotesovec, Mar 25 2010

A172215 Number of ways to place 6 nonattacking knights on a 6 X n board.

Original entry on oeis.org

1, 58, 729, 8830, 60285, 257318, 858262, 2404448, 5879329, 12927182, 26115008, 49238436, 87675623, 148787822, 242366502, 381127124, 581249573, 862965246, 1251190796, 1776208532, 2474393475, 3388987070, 4570917554, 6079666980

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A061992, A172212, A172213, A172214.

CoefficientList[Series[-(104 x^15 - 116 x^14 - 1328 x^13 + 3992 x^12 + 806 x^11 - 16380 x^10 + 27343 x^9 - 4845 x^8 - 15537 x^7 + 38275 x^6 - 2753 x^5 + 11789 x^4 + 4910 x^3 + 344 x^2 + 51 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,58,729,8830,60285,257318,858262,2404448,5879329,12927182,26115008,49238436,87675623,148787822,242366502,381127124},30] (* Harvey P. Dale, Dec 31 2022 *)

a(n) = (648n^6-11340n^5+103770n^4-606645n^3+2328317n^2-5466660n+6051720)/10, n>=10.
G.f.: -x * (104*x^15 -116*x^14 -1328*x^13 +3992*x^12 +806*x^11 -16380*x^10 +27343*x^9 -4845*x^8 -15537*x^7 +38275*x^6 -2753*x^5 +11789*x^4 +4910*x^3 +344*x^2 +51*x +1) / (x-1)^7. - Vaclav Kotesovec, Mar 25 2010
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). - Wesley Ivan Hurt, Apr 16 2023

A172217 Number of ways to place 7 nonattacking knights on a 7 X n board.

Original entry on oeis.org

1, 78, 1758, 38588, 383246, 2135344, 8891854, 30108310, 86669806, 219845764, 504261973, 1065642840, 2104251027, 3924818982, 6973786593, 11884673662, 19532410762, 31097451768, 48140491605, 72688612756, 107333684073

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A061993, A172212, A172213, A172214, A172215.

CoefficientList[Series[(252 x^18 - 272 x^17 - 5134 x^16 + 14468 x^15 + 19721 x^14 - 132666 x^13 + 174233 x^12 + 119440 x^11 - 540473 x^10 + 654954 x^9 - 89133 x^8 - 93778 x^7 + 497782 x^6 + 56796 x^5 + 119468 x^4 + 26652 x^3 + 1162 x^2 + 70 x + 1) / (x - 1)^8, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (117649n^7-2571471n^6+29223943n^5-216954465n^4+1114503256n^3-3907492824n^2+8562799512n-8962924320)/720,n>=12.
For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - 3(k-1)(3k-4)/2/k!*(kn)^(k-1) + ...
G.f.: x * (252*x^18 -272*x^17 -5134*x^16 +14468*x^15 +19721*x^14 -132666*x^13 +174233*x^12 +119440*x^11 -540473*x^10 +654954*x^9 -89133*x^8 -93778*x^7 +497782*x^6 +56796*x^5 +119468*x^4 +26652*x^3 +1162*x^2 +70*x +1) / (x-1)^8. - Vaclav Kotesovec, Mar 25 2010

A172219 Number of ways to place 4 nonattacking nightriders on a 4 X n board.

Original entry on oeis.org

1, 16, 84, 412, 1126, 2760, 5739, 10982, 19695, 33068, 52801, 80638, 118731, 169368, 235135, 318890, 423733, 553028, 710389, 899690, 1125059, 1390880, 1701793, 2062694, 2478735, 2955324, 3498125, 4113058, 4806299, 5584280, 6453689

Offset: 1

Vaclav Kotesovec, Jan 29 2010

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A061990, A172213, A172218.

CoefficientList[Series[-(2 x^21 - 6 x^20 + 10 x^19 - 14 x^18 + 22 x^17 - 30 x^16 - 26 x^15 + 162 x^14 - 272 x^13 + 364 x^12 - 466 x^11 + 526 x^10 - 303 x^9 - 207 x^8 + 603 x^7 - 517 x^6 + 489 x^5 - 249 x^4 + 142 x^3 + 14 x^2 + 11 x + 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (32n^4 - 432n^3 + 3190n^2 - 13323n + 25530)/3, n>=18.

G.f.: -x * (2*x^21 -6*x^20 +10*x^19 -14*x^18 +22*x^17 -30*x^16 -26*x^15 +162*x^14 -272*x^13 +364*x^12 -466*x^11 +526*x^10 -303*x^9 -207*x^8 +603*x^7 -517*x^6 +489*x^5 -249*x^4 +142*x^3 +14*x^2 +11*x +1) / (x-1)^5. - Vaclav Kotesovec, Mar 25 2010

A174698 Number of ways to place 8 nonattacking knights on an 8 X n board.

Original entry on oeis.org

1, 81, 4409, 175720, 2479881, 17925691, 92952858, 379978716, 1286959255, 3765248749, 9805497200, 23226916560, 50866495373, 104288896551, 202154535834, 373400685738, 661407061211, 1129334088897, 1866838857216

Offset: 1

Vaclav Kotesovec, Mar 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A172449, A172212, A172213, A172214, A172215, A172217.

CoefficientList[Series[(592 x^21 - 584 x^20 - 18100 x^19 + 49628 x^18 + 134264 x^17 - 735838 x^16 + 584418 x^15 + 2607764 x^14 - 7093608 x^13 + 5656936 x^12 + 5136811 x^11 - 13973779 x^10 + 14583702 x^9 - 1612610 x^8 + 2009820 x^7 + 6682287 x^6 + 1572406 x^5 + 1050447 x^4 + 138871 x^3 + 3716 x^2 + 72 x + 1) / (1 - x)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

Explicit formula: a(n) = (262144*n^8 -6881280*n^7+93456384*n^6 -838693632*n^5 +5361604836*n^4 -24739168020*n^3+79766188151*n^2 -163079018193*n +160750559340)/630, n>=14.

G.f.: x*(592*x^21 -584*x^20 -18100*x^19 +49628*x^18+134264*x^17 -735838*x^16 +584418*x^15+2607764*x^14 -7093608*x^13 +5656936*x^12 +5136811*x^11 -13973779*x^10 +14583702*x^9 -1612610*x^8 +2009820*x^7 +6682287*x^6 +1572406*x^5 +1050447*x^4 +138871*x^3+3716*x^2 +72*x+1)/(1-x)^9.

cp's OEIS Frontend

A172214 Number of ways to place 5 nonattacking knights on a 5 X n board.

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A172215 Number of ways to place 6 nonattacking knights on a 6 X n board.

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A172217 Number of ways to place 7 nonattacking knights on a 7 X n board.

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A172219 Number of ways to place 4 nonattacking nightriders on a 4 X n board.

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A174698 Number of ways to place 8 nonattacking knights on an 8 X n board.

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