A333257 Number of distinct consecutive subsequence-sums of the k-th composition in standard order.

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 5, 4, 5, 5, 5, 2, 4, 4, 6, 4, 6, 5, 6, 4, 5, 6, 6, 6, 6, 6, 6, 2, 4, 4, 6, 3, 6, 6, 7, 4, 7, 4, 7, 6, 7, 6, 7, 4, 5, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 2, 4, 4, 6, 4, 7, 7, 8, 4, 6, 6, 8, 5, 7, 7, 8, 4, 7, 5, 8, 6, 8, 7

Offset

0,2

Comments

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.

Links

Table of n, a(n) for n=0..86.

Formula

a(n) = A333224(n) + 1.

Example

The ninth composition in standard order is (3,1), which has consecutive subsequences (), (1), (3), (3,1), with sums 0, 1, 3, 4, so a(9) = 4.

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n], {___, s___, ___}:>Plus[s]]]], {n, 0, 100}]

Crossrefs

Dominated by A124771.

Compositions where every subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222, while the case of partitions is counted by A325768 and ranked by A325779.

Positive subset-sums of partitions are counted by A276024 and A299701.

Knapsack partitions are counted by A108917 and ranked by A299702.

Knapsack compositions are counted by A325676 and A325687 and ranked by A333223.

The version for Heinz numbers of partitions is A325770.

Not allowing empty subsequences gives A333224.

Cf. A000120, A029931, A048793, A059519, A066099, A070939, A114994, A124765, A124767, A233564, A272919, A325778, A333217.

Sequence in context: A049822 A140060 A164341 * A334968 A124771 A334299

Adjacent sequences: A333254 A333255 A333256 * A333258 A333259 A333260

Keyword

nonn

Author

Gus Wiseman, Mar 20 2020

Status

approved