|
|
A345919
|
|
Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum < 0.
|
|
28
|
|
|
6, 12, 20, 24, 25, 27, 30, 40, 48, 49, 51, 54, 60, 72, 80, 81, 83, 86, 92, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 111, 116, 120, 121, 123, 126, 144, 160, 161, 163, 166, 172, 184, 192, 193, 194, 195, 197, 198, 199, 202, 204, 205, 207, 212, 216, 217, 219
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
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
|
|
|
EXAMPLE
|
The initial terms and the corresponding compositions:
6: (1,2) 81: (2,4,1)
12: (1,3) 83: (2,3,1,1)
20: (2,3) 86: (2,2,1,2)
24: (1,4) 92: (2,1,1,3)
25: (1,3,1) 96: (1,6)
27: (1,2,1,1) 97: (1,5,1)
30: (1,1,1,2) 98: (1,4,2)
40: (2,4) 99: (1,4,1,1)
48: (1,5) 101: (1,3,2,1)
49: (1,4,1) 102: (1,3,1,2)
51: (1,3,1,1) 103: (1,3,1,1,1)
54: (1,2,1,2) 106: (1,2,2,2)
60: (1,1,1,3) 108: (1,2,1,3)
72: (3,4) 109: (1,2,1,2,1)
80: (2,5) 111: (1,2,1,1,1,1)
|
|
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], ats[stc[#]]<0&]
|
|
CROSSREFS
|
The version for Heinz numbers of partitions is A119899.
These are the positions of terms < 0 in A124754.
The weak (k <= 0) version is A345915.
The opposite (k < 0) version is A345917.
The version for reversed alternating sum is A345920.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
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/reverse-alternating sum k:
Cf. A000070, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|