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A351010
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Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).
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7
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0, 3, 10, 15, 36, 43, 58, 63, 136, 147, 170, 175, 228, 235, 250, 255, 528, 547, 586, 591, 676, 683, 698, 703, 904, 915, 938, 943, 996, 1003, 1018, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2362, 2367, 2696, 2707, 2730, 2735, 2788, 2795, 2810, 2815, 3600, 3619
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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.
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LINKS
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EXAMPLE
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The terms together with their binary expansions and the corresponding compositions begin:
0: 0 ()
3: 11 (1,1)
10: 1010 (2,2)
15: 1111 (1,1,1,1)
36: 100100 (3,3)
43: 101011 (2,2,1,1)
58: 111010 (1,1,2,2)
63: 111111 (1,1,1,1,1,1)
136: 10001000 (4,4)
147: 10010011 (3,3,1,1)
170: 10101010 (2,2,2,2)
175: 10101111 (2,2,1,1,1,1)
228: 11100100 (1,1,3,3)
235: 11101011 (1,1,2,2,1,1)
250: 11111010 (1,1,1,1,2,2)
255: 11111111 (1,1,1,1,1,1,1,1)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], And@@EvenQ/@Length/@Split[stc[#]]&]
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CROSSREFS
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The case of twins (binary weight 2) is A000120.
The Heinz numbers of these compositions are given by A000290.
All terms are evil numbers A001969.
The strict case (distinct twins) is A351009, counted by A032020 with 0's.
The anti-run case is A351011, counted by A003242 interspersed with 0's.
A011782 counts integer compositions.
A351014 counts distinct runs in standard compositions.
Cf. A018819, A025047, A027383, A035457, A053738, A088218, A106356, A238279, A344604, A351012, A351015.
Selected statistics of standard compositions:
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Constant compositions are A272919.
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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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