OFFSET
1,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n. 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.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
LINKS
EXAMPLE
The sequence together with the corresponding compositions begins:
0: ()
1: (1)
9: (3,1)
19: (3,1,1)
22: (2,1,2)
28: (1,1,3)
34: (4,2)
69: (4,2,1)
74: (3,2,2)
84: (2,2,3)
104: (1,2,4)
132: (5,3)
135: (5,1,1,1)
141: (4,1,2,1)
153: (3,1,3,1)
177: (2,1,4,1)
225: (1,1,5,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Select[Range[0, 100], Length[stc[#]]==ats[stc[#]]&]
CROSSREFS
See link for sequences related to standard compositions.
These compositions are counted by A357182.
The case of partitions is counted by A357189.
A011782 counts compositions.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 28 2022
STATUS
approved