login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A116694 Array read by antidiagonals: number of ways of dividing an n X m rectangle into integer sided rectangles. 17
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Antidiagonals n = 1..20, flattened

Joshua Smith and Helena Verrill, On dividing rectangles into rectangles

EXAMPLE

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, ...

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-Fran├žois Alcover, Feb 23 2015, after Alois P. Heinz *)

PROG

(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<m&&!A[x+1, Y]&&for(j=Y+1, N, for(i=X, x, A[i, j]=0))); L}

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.

Sequence in context: A300182 A317532 A222659 * A220810 A221024 A220545

Adjacent sequences:  A116691 A116692 A116693 * A116695 A116696 A116697

KEYWORD

nonn,tabl

AUTHOR

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

EXTENSIONS

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

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 November 30 11:33 EST 2020. Contains 338799 sequences. (Running on oeis4.)