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 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. 10
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified April 21 17:36 EDT 2021. Contains 343156 sequences. (Running on oeis4.)