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A345914
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Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.
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25
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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
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OFFSET
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1,3
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COMMENTS
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The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) 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.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Select[Range[0, 100], sats[stc[#]]>=0&]
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CROSSREFS
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These compositions are counted by A116406.
The case of non-Heinz 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 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).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
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, A008549, A025047, A027187, A032443, A034871, A114121, A120452, A163493, A238279, A344650, A344743.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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