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.

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A116694 Array read by antidiagonals: number of ways of dividing an n X m rectangle into integer-sided rectangles.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 34, 34, 8, 16, 148, 322, 148, 16, 32, 650, 3164, 3164, 650, 32, 64, 2864, 31484, 70878, 31484, 2864, 64, 128, 12634, 314662, 1613060, 1613060, 314662, 12634, 128, 256, 55756, 3149674, 36911922, 84231996, 36911922, 3149674, 55756, 256
Offset: 1

Views

Author

Helena Verrill (verrill(AT)math.lsu.edu), Feb 23 2006

Keywords

Examples

			Array begins:
   1,    2,      4,        8,         16,           32, ...
   2,    8,     34,      148,        650,         2864, ...
   4,   34,    322,     3164,      31484,       314662, ...
   8,  148,   3164,    70878,    1613060,     36911922, ...
  16,  650,  31484,  1613060,   84231996,   4427635270, ...
  32, 2864, 314662, 36911922, 4427635270, 535236230270, ...

Links

Crossrefs

Columns (or rows) 1-10 give: A011782, A034999, A208215, A220297, A220298, A220299, A220300, A220301, A220302, A220303.
Main diagonal gives A182275.
For irreducible or "tight" pavings, see also A285357.
Triangular version: A333476.
A(2n,n) gives A333495.

Programs

  • Maple
    M:= proc(n) option remember; local k; k:= 2^(n-2);
      `if`(n=1, Matrix([2]), Matrix(2*k, (i, j)->`if`(i<=k,
      `if`(j<=k, M(n-1)[i, j], B(n-1)[i, j-k]),
      `if`(j<=k, B(n-1)[i-k, j], 2*M(n-1)[i-k, j-k]))))
    end:
B:= proc(n) option remember; local k; k:=2^(n-2);
      `if`(n=1, Matrix([1]), Matrix(2*k, (i,j)->`if`(i<=k,
      `if`(j<=k, B(n-1)[i, j], B(n-1)[i, j-k]),
      `if`(j<=k, B(n-1)[i-k, j], M(n-1)[i-k, j-k]))))
    end:
A:= proc(n, m) option remember; `if`(n=0 or m=0, 1, `if`(m>n, A(m, n),
      add(i, i=map(rhs, [op(op(2, M(m)^(n-1)))]))))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Dec 13 2012
  • Mathematica
    M[n_] := M[n] = Module[{k = 2^(n-2)}, If[n == 1, {{2}}, Table[If[i <= k, If[j <= k, M[n-1][[i, j]], B[n-1][[i, j-k]]], If[j <= k, B[n-1][[i-k, j]], 2*M[n-1][[i-k, j-k]]]], {i, 1, 2k}, {j, 1, 2k}]]]; B[n_] := B[n] = Module[{k = 2^(n-2)}, If[n == 1, {{1}}, Table[If[i <= k, If[j <= k, B[n-1][[i, j]], B[n-1][[i, j-k]]], If[j <= k, B[n-1][[i-k, j]], M[n-1][[i-k, j-k]]]], {i, 1, 2k}, {j, 1, 2k}]]]; A[0, 0] = 1; A[n_ , m_ ] /; m>n := A[m, n]; A[n_ , m_ ] :=MatrixPower[M[m], n-1] // Flatten // Total; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Feb 23 2015, after Alois P. Heinz *)
  • PARI
    A116694(m,n)=#fill(m,n) \\ where fill() below computes all tilings. - M. F. Hasler, Jan 22 2018
fill(m,n,A=matrix(m,n),i=1,X=1,Y=1)={while((Y>n&&X++&&!Y=0)||A[X,Y], X>m&&return([A]); Y++); my(N=n,L=[]); for(x=X,m, A[x,Y]&&break; for(y=Y,N, if(A[x,y],for(j=y,N,for(k=X,x-1,A[k,j]=0));N=y-1;break); for(j=X,x,A[j,y]=i); L=concat(L,fill(m,n,A,i+1,X,y+1))); x

Extensions

Edited and more terms from Alois P. Heinz, Dec 09 2012

A333476 Triangle read by rows: T(n,k) gives the number of ways to partition an n X k grid into rectangles of integer side lengths with 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 1, 4, 34, 322, 1, 8, 148, 3164, 70878, 1, 16, 650, 31484, 1613060, 84231996, 1, 32, 2864, 314662, 36911922, 4427635270, 535236230270, 1, 64, 12634, 3149674, 846280548, 233276449488, 64878517290010, 18100579400986674
Offset: 0

Views

Author

Peter Kagey, Mar 23 2020

Keywords

Examples

			Triangle begins:
n\k| 0   1     2       3         4           5             6
---+--------------------------------------------------------
  0| 1;
  1| 1,  1;
  2| 1,  2,    8;
  3| 1,  4,   34,    322;
  4| 1,  8,  148,   3164,    70878;
  5| 1, 16,  650,  31484,  1613060,   84231996;
  6| 1, 32, 2864, 314662, 36911922, 4427635270, 535236230270;
     ...

Links

Crossrefs

Triangular version of A116694.
Columns 0-10 are given by: A000012, A011782, A034999, A208215, A220297, A220298, A220299, A220300, A220301, A220302, A220303.
Main diagonal is given by A182275.
T(2n,n) gives A333495.

Programs

  • Maple
    M:= proc(n) option remember; local k; k:= 2^(n-2);
      `if`(n=1, Matrix([2]), Matrix(2*k, (i, j)->`if`(i<=k,
      `if`(j<=k, M(n-1)[i, j], B(n-1)[i, j-k]),
      `if`(j<=k, B(n-1)[i-k, j], 2*M(n-1)[i-k, j-k]))))
    end:
B:= proc(n) option remember; local k; k:=2^(n-2);
      `if`(n=1, Matrix([1]), Matrix(2*k, (i, j)->`if`(i<=k,
      `if`(j<=k, B(n-1)[i, j], B(n-1)[i, j-k]),
      `if`(j<=k, B(n-1)[i-k, j], M(n-1)[i-k, j-k]))))
    end:
T:= proc(n, m) option remember; `if`((s-> 0 in s or s={1})(
      {n, m}), 1, `if`(m>n, T(m, n), add(i, i=map(rhs,
       [op(op(2, M(m)^(n-1)))]))))
    end:
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Mar 23 2020
  • Mathematica
    M[n_] := M[n] = Module[{k = 2^(n - 2)}, If[n == 1, {{2}}, Table[If[i <= k, If[j <= k, M[n - 1][[i, j]], B[n - 1][[i, j - k]]], If[j <= k, B[n - 1][[i - k, j]], 2 M[n - 1][[i - k, j - k]]]], {i, 1, 2k}, {j, 1, 2k}]]];
B[n_] := B[n] = Module[{k = 2^(n - 2)}, If[n == 1, {{1}}, Table[If[i <= k, If[j <= k, B[n - 1][[i, j]], B[n - 1][[i, j - k]]], If[j <= k, B[n - 1][[i - k, j]], M[n - 1][[i - k, j - k]]]], {i, 1, 2k}, {j, 1, 2k}]]];
T[_, 0] = 1;
T[n_, k_] /; k > n := T[k, n];
T[n_, k_] := MatrixPower[M[k], n-1] // Flatten // Total;
Table[Table[T[n, k], {k, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Formula

T(n,k) = A116694(n,k).
Showing 1-2 of 2 results.

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