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A345922
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Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum 2.
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28
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2, 11, 12, 14, 37, 40, 42, 47, 51, 52, 54, 59, 60, 62, 137, 144, 146, 151, 157, 163, 164, 166, 171, 172, 174, 181, 184, 186, 191, 197, 200, 202, 207, 211, 212, 214, 219, 220, 222, 229, 232, 234, 239, 243, 244, 246, 251, 252, 254, 529, 544, 546, 551, 557, 569
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OFFSET
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1,1
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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:
2: (2) 144: (3,5)
11: (2,1,1) 146: (3,3,2)
12: (1,3) 151: (3,2,1,1,1)
14: (1,1,2) 157: (3,1,1,2,1)
37: (3,2,1) 163: (2,4,1,1)
40: (2,4) 164: (2,3,3)
42: (2,2,2) 166: (2,3,1,2)
47: (2,1,1,1,1) 171: (2,2,2,1,1)
51: (1,3,1,1) 172: (2,2,1,3)
52: (1,2,3) 174: (2,2,1,1,2)
54: (1,2,1,2) 181: (2,1,2,2,1)
59: (1,1,2,1,1) 184: (2,1,1,4)
60: (1,1,1,3) 186: (2,1,1,2,2)
62: (1,1,1,1,2) 191: (2,1,1,1,1,1,1)
137: (4,3,1) 197: (1,4,2,1)
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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[#]]==2&]
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CROSSREFS
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These compositions are counted by A088218.
The case of partitions is counted by A120452.
These are the positions of 2's in A344618.
The opposite (negative 2) version is A345923.
The version for unreversed alternating sum is A345925.
The version for Heinz numbers of partitions is A345961.
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).
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, A000097, A025047, A027193, A034871, A114121, A163493, A236913, A344607, A344608, A344741, 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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