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A145111
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Square array A(n,k) of numbers of length n binary words with less than k 0-digits between any pair of consecutive 1-digits (n,k >= 0), read by antidiagonals.
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7
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1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 4, 7, 5, 1, 2, 4, 8, 11, 6, 1, 2, 4, 8, 15, 16, 7, 1, 2, 4, 8, 16, 27, 22, 8, 1, 2, 4, 8, 16, 31, 47, 29, 9, 1, 2, 4, 8, 16, 32, 59, 80, 37, 10, 1, 2, 4, 8, 16, 32, 63, 111, 134, 46, 11, 1, 2, 4, 8, 16, 32, 64, 123, 207, 222, 56, 12, 1, 2, 4, 8, 16, 32, 64
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| A(n,k) = maximal number of regions into which k-space can be divided by n hyperplanes (k >= 1, n >= 0). For all fixed k, the sequences A(n,k) are complete. [Frank M Jackson, Jan 26 2012]
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LINKS
| Alois P. Heinz, antidiagonals n = 0..140
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FORMULA
| G.f. of column k: (1-x+x^(k+1))/(1-3*x+2*x^2+x^(k+1)-x^(k+2)).
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EXAMPLE
| A (4,1) = 11, because 11 binary words of length 4 have less than 1 0-digit between any pair of consecutive 1-digits: 0000, 0001, 0010, 0100, 1000, 0011, 0110, 1100, 0111, 1110, 1111.
Square array A(n,k) begins:
1 1 1 1 1 1 ...
2 2 2 2 2 2 ...
3 4 4 4 4 4 ...
4 7 8 8 8 8 ...
5 11 15 16 16 16 ...
6 16 27 31 32 32 ...
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MAPLE
| f:= proc(n, k) option remember; local j; if n=0 then 1 elif n<=k then 2^(n-1) else add (f(n-j, k), j=1..k) fi end: g:= proc(n, k) option remember; if n<0 then 0 else g(n-1, k) +f(n, k) fi end: A:= (n, k)-> `if`(n=0, g(0, k), A(n-1, k) +g(n-1, k)): seq (seq (A(n, d-n), n=0..d), d=0..14);
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CROSSREFS
| Columns 0-9 give: A000027(n+1), A000124, A000126(n+1), A007800(n+1), A145112, A145113, A145114, A145115, A145116, A145117. Diagonal gives: A000079. See also: A141539.
Sequence in context: A195076 A163491 A080772 * A104795 A116925 A178030
Adjacent sequences: A145108 A145109 A145110 * A145112 A145113 A145114
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KEYWORD
| nonn,tabl
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AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 02 2008
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