The On-Line Encyclopedia of Integer Sequences (OEIS)

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A359043 revision #9

A359043

Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.

0, 1, 2, 2, 3, 4, 3, 3, 4, 6, 5, 6, 4, 5, 4, 4, 5, 8, 7, 9, 6, 8, 7, 8, 5, 7, 6, 7, 5, 6, 5, 5, 6, 10, 9, 12, 8, 11, 10, 12, 7, 10, 9, 11, 8, 10, 9, 10, 6, 9, 8, 10, 7, 9, 8, 9, 6, 8, 7, 8, 6, 7, 6, 6, 7, 12, 11, 15, 10, 14, 13, 16, 9, 13, 12, 15, 11, 14, 13

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OFFSET

0,3

COMMENTS

We define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts.

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.

LINKS

Table of n, a(n) for n=0..78.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

EXAMPLE

The 29th composition in standard order is (1,1,2,1), with adjusted partial sums (1,1,2,2), with sum 6, so a(29) = 6.

MATHEMATICA

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

Table[Total[Accumulate[stc[n]-1]+1], {n, 0, 100}]

CROSSREFS

See link for sequences related to standard compositions.

The unadjusted reverse version is A029931, row sums of A048793.

The reverse version is A161511, row sums of A125106.