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A357187
First differences A357186 = "Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything."
7
1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 1, 0, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -3, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0
OFFSET
0,32
COMMENTS
Are there any terms > 1?
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
FORMULA
a(n) = A357186(n + 1) - A357186(n).
EXAMPLE
We have A357186(5) - A357186(4) = 3 - 2 = 1, so a(4) = 1.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Differences[Table[stc/@stc[n]/.List->Plus, {n, 0, 100}]]
CROSSREFS
See link for sequences related to standard compositions.
Positions of first appearances appear to all belong to A052955.
Differences of A357186 (row-sums of A357135).
The version for partitions is A357458, differences of A325033.
Sequence in context: A090584 A171400 A271592 * A128409 A133699 A157361
KEYWORD
sign
AUTHOR
Gus Wiseman, Sep 28 2022
STATUS
approved