

A345917


Numbers k such that the kth composition in standard order (row k of A066099) has alternating sum > 0.


31



1, 2, 4, 5, 7, 8, 9, 11, 14, 16, 17, 18, 19, 21, 22, 23, 26, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 42, 44, 45, 47, 52, 56, 57, 59, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 84, 85, 87, 88, 89, 90, 91, 93, 94, 95, 100, 104, 105, 107
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (1)^(i1) y_i.
The kth composition in standard order (graded reverselexicographic, 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



EXAMPLE

The initial terms and the corresponding compositions:
1: (1)
2: (2)
4: (3)
5: (2,1)
7: (1,1,1)
8: (4)
9: (3,1)
11: (2,1,1)
14: (1,1,2)
16: (5)
17: (4,1)
18: (3,2)
19: (3,1,1)
21: (2,2,1)
22: (2,1,2)


MATHEMATICA

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
ats[y_]:=Sum[(1)^(i1)*y[[i]], {i, Length[y]}];
Select[Range[0, 100], ats[stc[#]]>0&]


CROSSREFS

The version for Heinz numbers of partitions is A026424.
These compositions are counted by A027306.
These are the positions of terms > 0 in A124754.
The weak (k >= 0) version is A345913.
The reversealternating version is A345918.
The opposite (k < 0) version is A345919.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A097805 counts compositions by alternating (or reversealternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Compositions of n, 2n, or 2n+1 with alternating/reversealternating sum k:
Cf. A000070, A000346, A008549, A025047, A027187, A027193, A032443, A114121, A163493, A344609, A345908.


KEYWORD

nonn


AUTHOR



STATUS

approved



