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A227736
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Irregular table read by rows: the first entry of n-th row is length of run of rightmost identical bits (either 0 or 1, equal to n mod 2), followed by length of the next run of bits, etc., in the binary representation of n, when scanned from the least significant to the most significant end.
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22
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1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 3, 4, 4, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 1, 3, 1, 4, 5, 5, 1, 1, 4, 1, 1, 1, 3, 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) terms. In rows 2^(k-1)..2^k-1 we have all the compositions (ordered partitions) of k. Other orderings of compositions: A101211, A066099, A108244 and A124734.
Each row n (n>=1) contains the initial A005811(n) nonzero terms from the beginning of row n of A227186. A070939(n) gives the sum of terms on row n, while A167489(n) gives the product of its terms. A136480 gives the first column. A101211 lists the terms of each row in reverse order.
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LINKS
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FORMULA
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EXAMPLE
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Table begins as:
Row n in Terms on
n binary that row
1 1 1;
2 10 1,1;
3 11 2;
4 100 2,1;
5 101 1,1,1;
6 110 1,2;
7 111 3;
8 1000 3,1;
9 1001 1,2,1;
10 1010 1,1,1,1;
11 1011 2,1,1;
12 1100 2,2;
13 1101 1,1,2;
14 1110 1,3;
15 1111 4;
16 10000 4,1;
etc. with the terms of row n appearing in reverse order compared how the runs of the same length appear in the binary expansion of n (Cf. A101211).
Illustration of initial terms:
---------------------------------------
k m Diagram Composition
---------------------------------------
. _
1 1 |_|_ 1;
2 1 |_| | 1, 1,
2 2 |_ _|_ 2;
3 1 |_ | | 2, 1,
3 2 |_|_| | 1, 1, 1,
3 3 |_| | 1, 2,
3 4 |_ _ _|_ 3;
4 1 |_ | | 3, 1,
4 2 |_|_ | | 1, 2, 1,
4 3 |_| | | | 1, 1, 1, 1,
4 4 |_ _|_| | 2, 1, 1,
4 5 |_ | | 2, 2,
4 6 |_|_| | 1, 1, 2,
4 7 |_| | 1, 3,
4 8 |_ _ _ _|_ 4;
5 1 |_ | | 4, 1,
5 2 |_|_ | | 1, 3, 1,
5 3 |_| | | | 1, 1, 2, 1,
5 4 |_ _|_ | | 2, 2, 1,
5 5 |_ | | | | 2, 1, 1, 1,
5 6 |_|_| | | | 1, 1, 1, 1, 1,
5 7 |_| | | | 1, 2, 1, 1,
5 8 |_ _ _|_| | 3, 1, 1,
5 9 |_ | | 3, 2,
5 10 |_|_ | | 1, 2, 2,
5 11 |_| | | | 1, 1, 1, 2,
5 12 |_ _|_| | 2, 1, 2,
5 13 |_ | | 2, 3,
5 14 |_|_| | 1, 1, 3,
5 15 |_| | 1, 4,
5 16 |_ _ _ _ _| 5;
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Also irregular triangle read by rows in which row k lists the compositions of k, k >= 1.
Triangle begins:
[1];
[1,1], [2];
[2,1], [1,1,1], [1,2],[3];
[3,1], [1,2,1], [1,1,1,1], [2,1,1], [2,2], [1,1,2], [1,3], [4];
[4,1], [1,3,1], [1,1,2,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], [1,2,1,1], [3,1,1], [3,2], [1,2,2], [1,1,1,2], [2,1,2], [2,3], [1,1,3], [1,4], [5];
(End)
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MATHEMATICA
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Array[Length /@ Reverse@ Split@ IntegerDigits[#, 2] &, 34] // Flatten (* Michael De Vlieger, Dec 11 2020 *)
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PROG
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(Haskell)
import Data.List (group)
a227736 n k = a227736_tabf !! (n-1) !! (k-1)
a227736_row n = a227736_tabf !! (n-1)
a227736_tabf = map (map length . group) $ tail a030308_tabf
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CROSSREFS
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Cf. A227738 and also A227739 for similar table for unordered partitions.
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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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