login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A244081 Number T(n,k) of ways to place k nonattacking knights on an n X n board; triangle T(n,k), n>=0, k<=0<=A030978(n), read by rows. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

In other words, the n-th row gives the coefficients of the independence polynomial of the n X n knight graph. - Eric W. Weisstein, May 05 2017

LINKS

Alois P. Heinz, Rows n = 0..12, flattened

Eric Weisstein's World of Mathematics, Independence Polynomial

Eric Weisstein's World of Mathematics, Knight Graph

Eric Weisstein's World of Mathematics, Knights Problem

EXAMPLE

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

MAPLE

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 k<d then g:=subsop(2*d+1+k=f, g) fi;

         if k>2 then g:=subsop(  d-2+k=f, g) fi;

         if k<d-1 then g:=subsop(d+2+k=f, g) fi;

         expand(b(n, subsop(k=f, l)) +b(n, g)*x)

      fi

    end:

T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, [true$(n*3)])):

seq(T(n), n=0..7);

MATHEMATICA

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 *)

CROSSREFS

Columns k=0-6 give: A000012, A000290, A172132, A172134, A172135, A172136, A178499.

T(n,n) gives A201540.

Row sums give A141243.

Cf. A030978.

Sequence in context: A155675 A230207 A277949 * A279445 A217285 A212635

Adjacent sequences:  A244078 A244079 A244080 * A244082 A244083 A244084

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jun 19 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 15 00:30 EST 2019. Contains 329988 sequences. (Running on oeis4.)