A254414 Number A(n,k) of tilings of a k X n rectangle using polyominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 7, 4, 1, 1, 8, 29, 29, 8, 1, 1, 16, 124, 257, 124, 16, 1, 1, 32, 533, 2408, 2408, 533, 32, 1, 1, 64, 2293, 22873, 50128, 22873, 2293, 64, 1, 1, 128, 9866, 217969, 1064576, 1064576, 217969, 9866, 128, 1, 1, 256, 42451, 2078716, 22734496, 50796983, 22734496, 2078716, 42451, 256, 1

Offset

0,8

Comments

A polyomino of shape I is a rectangle of width 1.

All columns (or rows) are linear recurrences with constant coefficients. An upper bound on the order of the recurrence is A005683(k+2). This upper bound is exact for at least 1 <= k <= 10. - Andrew Howroyd, Dec 23 2019

Links

Andrew Howroyd, Table of n, a(n) for n = 0..495

Wikipedia, Polyomino

Example

Square array A(n,k) begins:
  1,  1,    1,      1,        1,          1,            1, ...
  1,  1,    2,      4,        8,         16,           32, ...
  1,  2,    7,     29,      124,        533,         2293, ...
  1,  4,   29,    257,     2408,      22873,       217969, ...
  1,  8,  124,   2408,    50128,    1064576,     22734496, ...
  1, 16,  533,  22873,  1064576,   50796983,   2441987149, ...
  1, 32, 2293, 217969, 22734496, 2441987149, 264719566561, ...

Prog

(PARI)
step(v, S)={vector(#v, i, sum(j=1, #v, v[j]*2^hammingweight(bitand(S[i], S[j]))))}
mkS(k)={apply(b->bitand(b, 2*b+1), [2^(k-1)..2^k-1])}
T(n, k)={if(k<2, if(k==0||n==0, 1, 2^(n-1)), my(S=mkS(k), v=vector(#S, i, i==1)); for(n=1, n, v=step(v, S)); vecsum(v))} \\ Andrew Howroyd, Dec 23 2019

Crossrefs

Columns (or rows) k=0-7 give: A000012, A011782, A052961, A254124, A254125, A254126, A254458, A254607.

Main diagonal gives: A254127.

Cf. A005683.

Sequence in context: A177254 A340910 A132311 * A199802 A297347 A342623

Adjacent sequences: A254411 A254412 A254413 * A254415 A254416 A254417

Keyword

nonn,tabl

Author

Alois P. Heinz, Jan 30 2015

Status

approved