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A349153
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Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.
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3
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0, 11, 12, 14, 133, 138, 143, 148, 155, 158, 160, 168, 179, 182, 188, 195, 198, 204, 208, 216, 227, 230, 236, 240, 248, 2057, 2066, 2071, 2077, 2084, 2091, 2094, 2101, 2106, 2111, 2120, 2131, 2134, 2140, 2149, 2154, 2159, 2164, 2171, 2174, 2192, 2211, 2214
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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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
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LINKS
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EXAMPLE
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The terms and corresponding compositions begin:
0: ()
11: (2,1,1)
12: (1,3)
14: (1,1,2)
133: (5,2,1)
138: (4,2,2)
143: (4,1,1,1,1)
148: (3,2,3)
155: (3,1,2,1,1)
158: (3,1,1,1,2)
160: (2,6)
168: (2,2,4)
179: (2,1,3,1,1)
182: (2,1,2,1,2)
188: (2,1,1,1,3)
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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, 1000], Total[stc[#]]==2*sats[stc[#]]&]
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CROSSREFS
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These compositions are counted by A262977 up to 0's.
The unreversed negative version is A349154.
A non-reverse unordered version is A349159, counted by A000712 up to 0's.
A003242 counts Carlitz compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A294175.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Heinz number is given by A333219.
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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