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A345918
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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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27
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1, 2, 4, 6, 7, 8, 11, 12, 14, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 37, 38, 40, 42, 44, 47, 48, 51, 52, 54, 56, 59, 60, 62, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88, 91, 92, 93, 94, 96, 99, 100, 101, 102, 104, 106, 107, 108
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OFFSET
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1,2
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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 initial terms and the corresponding compositions:
1: (1) 26: (1,2,2) 52: (1,2,3)
2: (2) 27: (1,2,1,1) 54: (1,2,1,2)
4: (3) 28: (1,1,3) 56: (1,1,4)
6: (1,2) 30: (1,1,1,2) 59: (1,1,2,1,1)
7: (1,1,1) 31: (1,1,1,1,1) 60: (1,1,1,3)
8: (4) 32: (6) 62: (1,1,1,1,2)
11: (2,1,1) 35: (4,1,1) 64: (7)
12: (1,3) 37: (3,2,1) 67: (5,1,1)
14: (1,1,2) 38: (3,1,2) 69: (4,2,1)
16: (5) 40: (2,4) 70: (4,1,2)
19: (3,1,1) 42: (2,2,2) 72: (3,4)
20: (2,3) 44: (2,1,3) 73: (3,3,1)
21: (2,2,1) 47: (2,1,1,1,1) 74: (3,2,2)
22: (2,1,2) 48: (1,5) 76: (3,1,3)
24: (1,4) 51: (1,3,1,1) 79: (3,1,1,1,1)
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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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The version for prime indices is A000037.
The version for Heinz numbers of partitions is A026424, counted by A027193.
These compositions are counted by A027306.
These are the positions of terms > 0 in A344618.
The weak (k >= 0) version is A345914.
The version for unreversed alternating sum is A345917.
The opposite (k < 0) version is A345920.
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, A000290, A000346, A008549, A025047, A027187, A028260, A034871, A114121, A163493, A344607, A344608, A344650, A344743, A345908.
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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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