cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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)).

Original entry on oeis.org

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
Offset: 0

Views

Author

Alois P. Heinz, Jun 04 2014

Keywords

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

Examples

			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, ...
  ...

Links

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.

Programs

  • 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`(k1 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 (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[k1 && 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 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-François Alcover, Jun 09 2018, after Alois P. Heinz *)

A239273 Number of domicule tilings of a 2n X 2n square grid.

Original entry on oeis.org

1, 3, 280, 3037561, 3263262629905, 326207195516663381931, 3011882198082438957330143630563, 2565014347691062208319404612723752103028288, 201442620359313683494245316355883565275531844406384955392, 1458834332808489549111708247664894524221330758005874053074138540424018259
Offset: 0

Views

Author

Alois P. Heinz, Mar 13 2014

Keywords

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.
Number of perfect matchings in the 2n X 2n kings graph. - Andrew Howroyd, Apr 07 2016

Examples

			a(1) = 3:
  +---+   +---+   +---+
  |o o|   |o o|   |o-o|
  || ||   | X |   |   |
  |o o|   |o o|   |o-o|
  +---+   +---+   +---+.
a(2) = 280:
  +-------+ +-------+ +-------+ +-------+ +-------+
  |o o o-o| |o o o-o| |o-o o-o| |o o o o| |o o-o o|
  | X     | | X     | |       | | X  | || | \   / |
  |o o o o| |o o o o| |o o o o| |o o o o| |o o o o|
  |   /  || |   / / | ||  X  || |       | ||     ||
  |o o o o| |o o o o| |o o o o| |o-o o o| |o o o o|
  ||    \ | ||     || |       | |     X | | / /   |
  |o o-o o| |o o-o o| |o-o o-o| |o-o o o| |o o o-o|
  +-------+ +-------+ +-------+ +-------+ +-------+ ...

Links

Crossrefs

Even bisection of main diagonal of A239264.
Cf. A004003, A243510, A243424, A220638.

Programs

  • Mathematica
    b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, f = False, k}, Which[n == 0, 1, l[[1 ;; d]] == Array[f &, d], b[n - 1, Join[l[[d + 1 ;; 2*d]], Array[True &, d]]], True, For[k = 1, ! l[[k]], k++]; If[k < d && n > 1 && l[[k + d + 1]], b[n, ReplacePart[l, {k -> f, k + d + 1 -> f}]], 0] + If[k > 1 && n > 1 && l[[k + d - 1]], b[n, ReplacePart[l, {k -> f, k + d - 1 -> f}]], 0] + If[n > 1 && l[[k + d]], b[n, ReplacePart[l, {k -> f, k + d -> f}]], 0] + If[k < d && l[[k + 1]], b[n, ReplacePart[l, {k -> f, k + 1 -> f}]], 0]]];
A[n_, k_] := If[Mod[n*k, 2]>0, 0, If[k>n, A[k, n], b[n, Array[True&, k*2]]]];
a[n_] := A[2n, 2n];
Table[Print[n]; a[n], {n, 0, 7}] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A239264 *)

Formula

a(n) = A239264(2n,2n).

Extensions

a(8) from Alois P. Heinz, Sep 30 2014
a(9) from Alois P. Heinz, Nov 23 2018

A243511 Number of ways half the maximal number of domicules can be placed on an n X n square.

Original entry on oeis.org

1, 1, 6, 110, 18343, 9382388, 43019044258, 558605693874364, 66236678406528448449, 21324218858696872235790649, 63694548993039561351614359349824, 503614181552476916158019971522161589536, 37393123107750749849065997795023579513072683993, 7226766277015485498828489748946241590630740791829515424
Offset: 0

Views

Author

Alois P. Heinz, Jun 05 2014

Keywords

Examples

			a(2) = 6:
+---+  +---+  +---+  +---+  +---+  +---+
|o-o|  |   |  |o  |  |  o|  |o  |  |  o|
|   |  |   |  ||  |  |  ||  | \ |  | / |
|   |  |o-o|  |o  |  |  o|  |  o|  |o  |
+---+  +---+  +---+  +---+  +---+  +---+.

Crossrefs

Cf. A243510.

Formula

a(n) = A243424(n,floor(n^2/4)).
Showing 1-3 of 3 results.

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