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A227739
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Irregular table where row n lists in nondecreasing order the parts of unordered partition encoded in the runlengths of binary expansion of n; nonzero terms of A227189.
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15
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1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 3, 3, 3, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 2, 1, 3, 4, 4, 4, 1, 3, 3, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 2, 4, 1, 1, 3, 1, 4, 5, 5, 5, 1, 4, 4, 1, 1
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OFFSET
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1,4
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COMMENTS
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Row n has A005811(n) elements. Each row contains a unique (unordered) partition of some integer, and all possible partitions of finite natural numbers eventually occur. The first partition that sums to k occurs at row A227368(k) and the last at row A000225(k).
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LINKS
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FORMULA
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EXAMPLE
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Rows are constructed as:
Row n in Runlengths With one Partial sums The row sums
n binary collected subtracted of which give to, i.e. is
from lsb- from all terms on a partition of
to msb-end except 1st that row of A227183(n)
1 "1" [1] [1] 1; 1
2 "10" [1,1] [1,0] 1, 1; 2
3 "11" [2] [2] 2; 2
4 "100" [2,1] [2,0] 2, 2; 4
5 "101" [1,1,1] [1,0,0] 1, 1, 1; 3
6 "110" [1,2] [1,1] 1, 2; 3
7 "111" [3] [3] 3; 3
8 "1000" [3,1] [3,0] 3, 3; 6
9 "1001" [1,2,1] [1,1,0] 1, 2, 2; 5
10 "1010" [1,1,1,1] [1,0,0,0] 1, 1, 1, 1; 4
11 "1011" [2,1,1] [2,0,0] 2, 2, 2; 6
12 "1100" [2,2] [2,1] 2, 3; 5
13 "1101" [1,1,2] [1,0,1] 1, 1, 2; 4
14 "1110" [1,3] [1,2] 1, 3; 4
15 "1111" [4] [4] 4; 4
16 "10000" [4,1] [4,0] 4, 4; 8
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MATHEMATICA
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Table[Function[b, Accumulate@ Prepend[If[Length@ b > 1, Rest[b] - 1, {}], First@ b]]@ Map[Length, Split@ Reverse@ IntegerDigits[n, 2]], {n, 34}] // Flatten (* Michael De Vlieger, May 09 2017 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,base,tabf
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AUTHOR
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STATUS
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approved
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