A172228
Number of ways to place 5 nonattacking wazirs on an n X n board.
Original entry on oeis.org
0, 0, 1, 304, 10741, 127960, 870589, 4197456, 16005187, 51439096, 145085447, 369074128, 863338777, 1883786680, 3875953561, 7583888944, 14206566327, 25617069208, 44663199283, 75572017136, 124485188701, 200156902936, 314851577749, 485484612496, 735056106571, 1094434774968
Offset: 1
-
CoefficientList[Series[x^2 (6 x^11 - 26 x^10 - 93 x^9 + 527 x^8 + 490 x^7 - 6710 x^6 + 13630 x^5 - 3954 x^4 - 26364 x^3 - 7452 x^2 - 293 x - 1) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)
A244081
Number T(n,k) of ways to place k nonattacking knights on an n X n board; triangle T(n,k), n>=0, 0<=k<=A030978(n), read by rows.
Original entry on oeis.org
1, 1, 1, 1, 4, 6, 4, 1, 1, 9, 28, 36, 18, 2, 1, 16, 96, 276, 412, 340, 170, 48, 6, 1, 25, 252, 1360, 4436, 9386, 13384, 12996, 8526, 3679, 994, 158, 15, 1, 1, 36, 550, 4752, 26133, 97580, 257318, 491140, 688946, 716804, 556274, 323476, 141969, 47684, 12488, 2560, 393, 40, 2
Offset: 0
T(4,8) = 6:
._______. ._______. ._______. ._______. ._______. ._______.
|_|o|_|o| |o|_|o|_| |o|o|o|o| |o|_|_|o| |o|_|o|o| |o|o|_|o|
|o|_|o|_| |_|o|_|o| |_|_|_|_| |o|_|_|o| |_|_|_|o| |o|_|_|_|
|_|o|_|o| |o|_|o|_| |_|_|_|_| |o|_|_|o| |o|_|_|_| |_|_|_|o|
|o|_|o|_| |_|o|_|o| |o|o|o|o| |o|_|_|o| |o|o|_|o| |o|_|o|o| .
.
Triangle T(n,k) begins:
1;
1, 1;
1, 4, 6, 4, 1;
1, 9, 28, 36, 18, 2;
1, 16, 96, 276, 412, 340, 170, 48, 6;
1, 25, 252, 1360, 4436, 9386, 13384, 12996, 8526, 3679, 994, 158, 15, 1;
...
As independence polynomials:
1
1 + x
1 + 4*x + 6*x^2 + 4*x^3 + x^4
1 + 9*x + 28*x^2 + 36*x^3 + 18*x^4 + 2*x^5
1 + 16*x + 96*x^2 + 276*x^3 + 412*x^4 + 340*x^5 + 170*x^6 + 48*x^7 + 6*x^8
...
-
b:= proc(n, l) option remember; local d, f, g, k;
d:= nops(l)/3; f:=false;
if n=0 then 1
elif l[1..d]=[f$d] then b(n-1, [l[d+1..3*d][], true$d])
else for k while not l[k] do od; g:= subsop(k=f, l);
if k>1 then g:=subsop(2*d-1+k=f, g) fi;
if k2 then g:=subsop( d-2+k=f, g) fi;
if k(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, [true$(n*3)])):
seq(T(n), n=0..7);
-
b[n_, l_] := b[n, l] = Module[{d, f, g, k}, d = Length[l]/3; f = False; Which[n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n - 1, Join[l[[d + 1 ;; 3*d]], Array[True&, d]]], True, For[k = 1, !l[[k]], k++]; g = ReplacePart[l, k -> f];
If [k > 1, g = ReplacePart[g, 2*d - 1 + k -> f]];
If [k < d, g = ReplacePart[g, 2*d + 1 + k -> f]];
If [k > 2, g = ReplacePart[ g, d - 2 + k -> f]];
If [k < d - 1, g = ReplacePart[g, d + 2 + k -> f]];
Expand[b[n, ReplacePart[l, k -> f]] + b[n, g]*x]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][
b[n, Array[True&, n*3]]];
Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 28 2016, after Alois P. Heinz *)
Table[Count[IndependentVertexSetQ[KnightTourGraph[n, n], #] & /@ Subsets[Range[n^2], {k}], True], {n, 4}, {k, 0, If[n == 2, 4, (1 - (-1)^n + 2 n^2)/4]}] // Flatten (* Eric W. Weisstein, May 05 2017 *)
A201540
Number of ways to place n nonattacking knights on an n X n board.
Original entry on oeis.org
1, 6, 36, 412, 9386, 257318, 8891854, 379978716, 19206532478, 1120204619108, 74113608972922, 5483225594409823, 448414229054798028, 40154319792412218900, 3906519894750904583838
Offset: 1
-
b[n_, l_] := b[n, l] = Module[{d, f, g, k}, d = Length[l]/3; f = False; Which[n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n - 1, Join[l[[d + 1 ;; 3*d]], Array[True&, d]]], True, For[k = 1, ! l[[k]], k++]; g = ReplacePart[l, k -> f];
If[k > 1, g = ReplacePart[g, 2*d - 1 + k -> f]];
If[k < d, g = ReplacePart[g, 2*d + 1 + k -> f]];
If[k > 2, g = ReplacePart[g, d - 2 + k -> f]];
If[k < d - 1, g = ReplacePart[g, d + 2 + k -> f]];
Expand[b[n, ReplacePart[l, k -> f]] + b[n, g]*x]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Array[True&, n*3]]];
a[n_] := T[n][[n + 1]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Mar 29 2016, after Alois P. Heinz's code for A244081 *)
A172214
Number of ways to place 5 nonattacking knights on a 5 X n board.
Original entry on oeis.org
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
-
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 *)
A172140
Number of ways to place 5 nonattacking zebras on an n X n board.
Original entry on oeis.org
0, 0, 126, 2032, 20502, 160696, 929880, 4117520, 15037036, 47368960, 132577826, 336828368, 789558314, 1729320120, 3574328936, 7027309888, 13226773092, 23959787480, 41954706558, 71276149776, 117848892710, 190142197976
Offset: 1
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Vaclav 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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).
-
CoefficientList[Series[2x^2(100x^19 -648x^18 +1450x^17 -2126x^16 +10452x^15 - 43872x^14 +92798x^13 -100834x^12 +56460x^11 -61636x^10 +182288x^9 -303224x^8 + 275038x^7 -128982x^6 +21681x^5 +1933x^4 -13072x^3 -2540x^2 -323x-63)/(x-1)^11, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)
-
[0,0,126,2032,20502,160696,929880,4117520,15037036,47368960,132577826] + [(n^10 -90*n^8 +400*n^7 +2915*n^6 -26880*n^5 +2430*n^4 +609920*n^3 - 1517496*n^2 -4188480*n +16581120)/120 for n in (12..50)] # G. C. Greubel, Apr 19 2022
A172532
Number of ways to place 5 nonattacking knights on an n X n toroidal board.
Original entry on oeis.org
0, 0, 0, 128, 120, 30312, 283906, 1872064, 8643186, 31702920, 98179400, 267487920, 659015500, 1496908840, 3179369070, 6382030592, 12207535134, 22396355496, 39617305308, 67860021680
Offset: 1
-
CoefficientList[Series[- 2 x^3 (648 x^16 - 10328 x^15 + 71820 x^14 - 295572 x^13 + 818512 x^12 - 1640088 x^11 + 2492742 x^10 - 2967118 x^9 + 2825821 x^8 - 2185007 x^7 + 1376780 x^6 - 677852 x^5 + 219349 x^4 - 32023 x^3 + 18016 x^2 - 644 x + 64) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)
A243720
Number of inequivalent (mod D_8) ways to place 5 nonattacking knights on an n X n board.
Original entry on oeis.org
2, 49, 1244, 12329, 81900, 398907, 1562362, 5153001, 14907120, 38753358, 92417760, 204977323, 427812496, 847346181, 1604300270
Offset: 3
A178499
Number of ways to place 6 nonattacking knights on an n X n board.
Original entry on oeis.org
0, 0, 0, 170, 13384, 257318, 2774728, 20202298, 110018552, 481719518, 1781124856, 5756568738, 16676946372, 44127887910, 108192675468, 248568720338, 539925974784, 1116836380926, 2212958151968, 4220919779218
Offset: 1
-
CoefficientList[Series[- 2 x^3 (200 x^18 - 1540 x^17 + 2602 x^16 + 15442 x^15 - 98586 x^14 + 256698 x^13 - 336146 x^12 + 70977 x^11 + 587107 x^10 - 1302115 x^9 + 1569905 x^8 - 1100786 x^7 + 367130 x^6 - 212358 x^5 + 247682 x^4 + 212463 x^3 + 48293 x^2 + 5587 x + 85) / (x - 1)^13, {x, 0, 40}], x] (* Vincenzo Librandi, May 31 2013 *)
A172967
Number of ways to place 5 nonattacking knights on an n X n cylindrical board.
Original entry on oeis.org
0, 0, 0, 208, 3210, 58056, 458157, 2524176, 10587591, 36576380, 109008735, 289450344, 700477401, 1570789892, 3304892985, 6586928032, 12530769343, 22891446252
Offset: 1
-
CoefficientList[Series[- x^3 (468 x^16 - 7964 x^15 + 57164 x^14 - 238936 x^13 + 664383 x^12 - 1323653 x^11 + 1986964 x^10 - 2334676 x^9 + 2209082 x^8 - 1718662 x^7 + 1118210 x^6 - 595746 x^5 + 216519 x^4 - 38229 x^3 + 34186 x^2 + 922 x + 208) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)
Showing 1-9 of 9 results.
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