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A345921
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Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.
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26
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1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 44, 45, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81
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OFFSET
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1,2
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COMMENTS
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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.
Also numbers k such that the k-th composition in standard order has reverse-alternating sum != 0.
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LINKS
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EXAMPLE
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The initial terms and the corresponding compositions:
1: (1) 20: (2,3) 35: (4,1,1)
2: (2) 21: (2,2,1) 37: (3,2,1)
4: (3) 22: (2,1,2) 38: (3,1,2)
5: (2,1) 23: (2,1,1,1) 39: (3,1,1,1)
6: (1,2) 24: (1,4) 40: (2,4)
7: (1,1,1) 25: (1,3,1) 42: (2,2,2)
8: (4) 26: (1,2,2) 44: (2,1,3)
9: (3,1) 27: (1,2,1,1) 45: (2,1,2,1)
11: (2,1,1) 28: (1,1,3) 47: (2,1,1,1,1)
12: (1,3) 29: (1,1,2,1) 48: (1,5)
14: (1,1,2) 30: (1,1,1,2) 49: (1,4,1)
16: (5) 31: (1,1,1,1,1) 51: (1,3,1,1)
17: (4,1) 32: (6) 52: (1,2,3)
18: (3,2) 33: (5,1) 54: (1,2,1,2)
19: (3,1,1) 34: (4,2) 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;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Select[Range[0, 100], ats[stc[#]]!=0&]
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CROSSREFS
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The version for Heinz numbers of partitions is A000037.
These compositions are counted by A058622.
These are the positions of terms != 0 in A124754.
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).
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, A008549, A025047, A032443, A034871, A114121, A163493, A236913, A344609, A344651, 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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