A061991 -id:A061991 - cp's OEIS frontend

A172220 Number of ways to place 5 nonattacking nightriders on a 5 X n board.

1, 28, 157, 1248, 4650, 15162, 37988, 86958, 181423, 351708, 648441, 1127392, 1874194, 2988466, 4602096, 6870240, 9983347, 14163972, 19672403, 26812260, 35929480, 47418482, 61723238, 79341720, 100828175, 126796852, 157924785

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.

Vaclav Kotesovec and Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 40 terms from V. Kotesovec)
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A061991, A172214, A172218, A172219.

CoefficientList[Series[(2 x^36 - 8 x^35 + 16 x^34 - 24 x^33 + 38 x^32 - 64 x^31 + 104 x^30 - 156 x^29 + 54 x^28 + 380 x^27 - 944 x^26 + 1452 x^25 - 2172 x^24 + 3376 x^23 - 5094 x^22 + 7180 x^21 - 6614 x^20 - 28 x^19 + 8814 x^18 - 15212 x^17 + 21026 x^16 - 27284 x^15 + 34160 x^14 - 40598 x^13 + 39882 x^12 - 24490 x^11 + 3876 x^10 + 8558 x^9 - 11326 x^8 + 11266 x^7 -6006 x^6 + 3256 x^5 - 1028 x^4 + 706 x^3 + 4 x^2 + 22 x + 1) / (x - 1)^6, {x, 0, 40}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (625n^5-15250n^4+197915n^3-1588634n^2+7645896n-17283552)/24, n>=32.

G.f.: x * (2*x^36 -8*x^35 +16*x^34 -24*x^33 +38*x^32 -64*x^31 +104*x^30 -156*x^29 +54*x^28 +380*x^27 -944*x^26 +1452*x^25 -2172*x^24 +3376*x^23 -5094*x^22 +7180*x^21 -6614*x^20 -28*x^19 +8814*x^18 -15212*x^17 +21026*x^16 -27284*x^15 +34160*x^14 -40598*x^13 +39882*x^12 -24490*x^11 +3876*x^10 +8558*x^9 -11326*x^8 +11266*x^7 -6006*x^6 +3256*x^5 -1028*x^4 +706*x^3 +4*x^2 +22*x +1) / (x-1)^6. - Vaclav Kotesovec, Mar 25 2010

A269133 Number of ways to place m nonattacking queens on an m X n board, 1 <= m <= n (triangular array).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 6, 4, 2, 5, 12, 14, 12, 10, 6, 20, 36, 46, 40, 4, 7, 30, 76, 140, 164, 94, 40, 8, 42, 140, 344, 568, 550, 312, 92, 9, 56, 234, 732, 1614, 2292, 2038, 1066, 352, 10, 72, 364, 1400, 3916, 7552, 9632, 7828, 4040, 724, 11, 90, 536, 2468, 8492, 21362, 37248, 44148, 34774, 15116, 2680, 12, 110, 756, 4080, 16852, 52856, 120104, 195270, 222720, 160964, 68264, 14200

Offset: 1

Marko Riedel, Feb 19 2016

			The triangular array begins:
   n\m  1   2   3    4     5     6      7      8      9     10    11    12
   1    1
   2    2   0
   3    3   2   0
   4    4   6   4    2
   5    5  12  14   12    10
   6    6  20  36   46    40     4
   7    7  30  76  140   164    94     40
   8    8  42 140  344   568   550    312     92
   9    9  56 234  732  1614  2292   2038   1066    352
  10   10  72 364 1400  3916  7552   9632   7828   4040    724
  11   11  90 536 2468  8492 21362  37248  44148  34774  15116  2680
  12   12 110 756 4080 16852 52856 120104 195270 222720 160964 68264 14200
...

Math StackExchange, State space for eight queen problem
Marko Riedel, Perl program to compute triangular array of nonattacking queens configurations

Cf. A000170 (m=n), A002562, A065256, A348129.
Cf. A000027 (m=1), A002378 (m=2), A061989 (m=3), A061990 (m=4), A061991 (m=5), A061992 (m=6), A061993 (m=7), A172449 (m=8).
Cf. A036464 (2Q), A047659 (3Q), A061994 (4Q), A108792 (5Q), A176186 (6Q).
Cf. A006717, A051906, A319284 (backtrack trees).

{A269133(m, n, B=[], t=if(#B, setminus(n, Set(concat(B+t=[-#B..-1], B-t))), n=[1..n]))= if(#B < m-1, vecsum([A269133(m, setminus(n, [t]), concat(B,t)) | t<-t]), #t)} \\ M. F. Hasler, Jan 11 2022

cp's OEIS Frontend

A172220 Number of ways to place 5 nonattacking nightriders on a 5 X n board.

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