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A349799
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Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.
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12
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3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 47, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 79, 83, 84, 85, 86, 87, 90, 91, 94, 95, 99, 100, 103, 106, 111, 112, 113, 114, 115, 118, 119, 120, 121, 122, 123, 124, 125
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OFFSET
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1,1
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COMMENTS
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We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
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.
This sequence ranks compositions that are weakly but not strongly alternating.
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LINKS
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FORMULA
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EXAMPLE
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The terms and corresponding compositions begin:
3: (1,1)
7: (1,1,1)
10: (2,2)
11: (2,1,1)
14: (1,1,2)
15: (1,1,1,1)
19: (3,1,1)
21: (2,2,1)
23: (2,1,1,1)
26: (1,2,2)
27: (1,2,1,1)
28: (1,1,3)
29: (1,1,2,1)
30: (1,1,1,2)
31: (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;
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];
Select[Range[0, 100], (whkQ[stc[#]]||whkQ[-stc[#]])&&MatchQ[stc[#], {___, x_, x_, ___}]&]
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CROSSREFS
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Permutations of prime indices of this type are counted by A349798.
These compositions are counted by A349800.
A345164 = alternating permutations of prime indices, with twins A344606.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345166 = separable partitions with no alternations, ranked by A345173.
Cf. A005649, A049774, A096441, A128761, A344615, A345163, A349054, A349058, A349796, A349801, A350140.
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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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