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

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

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. 15
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.

Diagonal gives: A182275.

For irreducible or "tight" pavings, see also A285357.

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 22 20:30 EDT 2018. Contains 315270 sequences. (Running on oeis4.)