

A345914


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


25



0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

The reversealternating sum of a sequence (y_1,...,y_k) is Sum_i (1)^(ki) 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 sequence of terms together with the corresponding compositions begins:
0: () 19: (3,1,1) 40: (2,4)
1: (1) 20: (2,3) 41: (2,3,1)
2: (2) 21: (2,2,1) 42: (2,2,2)
3: (1,1) 22: (2,1,2) 43: (2,2,1,1)
4: (3) 24: (1,4) 44: (2,1,3)
6: (1,2) 26: (1,2,2) 46: (2,1,1,2)
7: (1,1,1) 27: (1,2,1,1) 47: (2,1,1,1,1)
8: (4) 28: (1,1,3) 48: (1,5)
10: (2,2) 30: (1,1,1,2) 50: (1,3,2)
11: (2,1,1) 31: (1,1,1,1,1) 51: (1,3,1,1)
12: (1,3) 32: (6) 52: (1,2,3)
13: (1,2,1) 35: (4,1,1) 53: (1,2,2,1)
14: (1,1,2) 36: (3,3) 54: (1,2,1,2)
15: (1,1,1,1) 37: (3,2,1) 55: (1,2,1,1,1)
16: (5) 38: (3,1,2) 56: (1,1,4)


MATHEMATICA

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


CROSSREFS

These compositions are counted by A116406.
The case of nonHeinz numbers of partitions is A119899, counted by A344608.
The version for Heinz numbers of partitions is A344609, counted by A344607.
These are the positions of terms >= 0 in A344618.
The version for unreversed alternating sum is A345913.
The opposite (k <= 0) version is A345916.
The strict (k > 0) case is A345918.
A097805 counts compositions by alternating (or reversealternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reversealternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reversealternating sum.
A344611 counts partitions of 2n with reversealternating sum >= 0.
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, A032443, A034871, A114121, A120452, A163493, A238279, A344650, A344743.


KEYWORD

nonn


AUTHOR



STATUS

approved



