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A243424 Triangle T(n,k) read by rows of number of ways k domicules can be placed on an n X n square (n >= 0, 0 <= k <= floor(n^2/2)). 9
1, 1, 1, 6, 3, 1, 20, 110, 180, 58, 1, 42, 657, 4890, 18343, 33792, 27380, 7416, 280, 1, 72, 2172, 36028, 362643, 2307376, 9382388, 24121696, 37965171, 34431880, 16172160, 3219364, 170985, 1, 110, 5375, 154434, 2911226, 38049764, 355340561, 2408715568 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

The n-th row gives the coefficients of the matching-generating polynomial of the n X n king graph. - Eric W. Weisstein, Jun 20 2017

LINKS

Alois P. Heinz, Rows n = 0..13, flattened

Eric Weisstein's World of Mathematics, King Graph

Eric Weisstein's World of Mathematics, Matching-Generating Polynomial

EXAMPLE

T(2,1) = 6:

  +---+  +---+  +---+  +---+  +---+  +---+

  |o-o|  |   |  |o  |  |  o|  |o  |  |  o|

  |   |  |   |  ||  |  |  ||  | \ |  | / |

  |   |  |o-o|  |o  |  |  o|  |  o|  |o  |

  +---+  +---+  +---+  +---+  +---+  +---+

T(2,2) = 3:

  +---+  +---+  +---+

  |o-o|  |o o|  |o o|

  |   |  || ||  | X |

  |o-o|  |o o|  |o o|

  +---+  +---+  +---+

Triangle T(n,k) begins:

  1;

  1;

  1,  6,    3;

  1, 20,  110,   180,     58;

  1, 42,  657,  4890,  18343,   33792,   27380,     7416,      280;

  1, 72, 2172, 36028, 362643, 2307376, 9382388, 24121696, 37965171, ...

  ...

MAPLE

b:= proc(n, l) option remember; local d, f, k;

      d:= nops(l)/2; f:=false;

      if n=0 then 1

    elif l[1..d]=[f$d] then b(n-1, [l[d+1..2*d][], true$d])

    else for k to d while not l[k] do od;

         expand(b(n, subsop(k=f, l))+

         `if`(k<d and n>1 and l[k+d+1],

                            x*b(n, subsop(k=f, k+d+1=f, l)), 0)+

         `if`(k>1 and n>1 and l[k+d-1],

                            x*b(n, subsop(k=f, k+d-1=f, l)), 0)+

         `if`(n>1 and l[k+d], x*b(n, subsop(k=f, k+d=f, l)), 0)+

         `if`(k<d and l[k+1], x*b(n, subsop(k=f, k+1=f, l)), 0))

      fi

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [true$(n*2)])):

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

MATHEMATICA

b[n_, l_] := b[n, l] = Module[{d, f, k}, d = Length[l]/2; f = False; Which[ n == 0, 1, l[[1 ;; d]] == Table[f, d], b[n-1, Join[l[[d+1 ;; 2d]], Table[ True, d]]], True, For[k = 1, !l[[k]], k++]; Expand[b[n, ReplacePart[l, k -> f]] + If[k<d && n>1 && l[[k+d+1]], x*b[n, ReplacePart[l, {k -> f, k + d + 1 -> f}]], 0] + If[k>1 && n>1 && l[[k + d - 1]], x*b[n, ReplacePart[ l, {k -> f, k + d - 1 -> f}]], 0] + If[n>1 && l[[k + d]], x*b[n, ReplacePart[l, {k -> f, k+d -> f}]], 0] + If[k<d && l[[k+1]], x*b[n, ReplacePart[l, {k -> f, k+1 -> f}]], 0]]]];

T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][

  b[n, Table[True, 2n]]];

Table[T[n], {n, 0, 7}] // Flatten (* Jean-Fran├žois Alcover, Jun 09 2018, after Alois P. Heinz *)

CROSSREFS

Columns k=0-5 give: A000012, A002943(n-1) for n>0, A243464, A243465, A243466, A243467.

Row sums give A220638.

T(n,floor(n^2/2)) gives A243510.

T(n,floor(n^2/4)) gives A243511.

Cf. A242861 (the same for dominoes), A239264.

Sequence in context: A240264 A119743 A272643 * A182227 A108451 A122178

Adjacent sequences:  A243421 A243422 A243423 * A243425 A243426 A243427

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jun 04 2014

STATUS

approved

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Last modified July 25 19:02 EDT 2021. Contains 346291 sequences. (Running on oeis4.)