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A349154
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Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.
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4
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0, 12, 160, 193, 195, 198, 204, 216, 240, 2304, 2561, 2563, 2566, 2572, 2584, 2608, 2656, 2752, 2944, 3074, 3077, 3079, 3082, 3085, 3087, 3092, 3097, 3099, 3102, 3112, 3121, 3123, 3126, 3132, 3152, 3169, 3171, 3174, 3180, 3192, 3232, 3265, 3267, 3270, 3276
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OFFSET
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1,2
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COMMENTS
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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.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
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LINKS
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EXAMPLE
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The terms and corresponding compositions begin:
0: ()
12: (1,3)
160: (2,6)
193: (1,6,1)
195: (1,5,1,1)
198: (1,4,1,2)
204: (1,3,1,3)
216: (1,2,1,4)
240: (1,1,1,5)
2304: (3,9)
2561: (2,9,1)
2563: (2,8,1,1)
2566: (2,7,1,2)
2572: (2,6,1,3)
2584: (2,5,1,4)
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MATHEMATICA
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ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 1000], Total[stc[#]]==-2*ats[stc[#]]&]
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CROSSREFS
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These compositions are counted by A224274 up to 0's.
A positive unordered version is A349159, counted by A000712 up to 0's.
A000346 = even-length compositions with alt sum != 0, complement A001700.
A003242 counts Carlitz compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
Classes of standard compositions:
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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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