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A298644
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The indices of the Carlitz compositions (i.e., compositions without adjacent equal parts).
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12
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1, 3, 4, 6, 7, 8, 9, 14, 15, 16, 17, 24, 27, 28, 30, 31, 32, 33, 35, 36, 39, 48, 49, 54, 55, 57, 59, 60, 62, 63, 64, 65, 67, 68, 70, 72, 73, 78, 79, 96, 97, 99, 110, 111, 112, 118, 119, 120, 121, 123, 124, 126, 127, 128, 129, 131, 132, 134, 135, 136, 137, 143, 144, 145, 156, 158
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OFFSET
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1,2
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COMMENTS
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We define the index of a composition to be the positive integer whose binary form has run-lengths (i.e., runs of 1's, runs of 0's, etc., from left to right) equal to the parts of the composition. Example: the composition [1,1,3,1] has index 46 since the binary form of 46 is 101110.
The command c(n) from the Maple program yields the composition having index n.
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LINKS
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EXAMPLE
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135 is in the sequence since its binary form is 10000111 and the composition [1,4,3] has no adjacent equal parts.
139 is not in the sequence since its binary form is 10001011 and the composition [1,3,1,1,2] has two adjacent equal parts.
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MAPLE
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Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: pd := proc (n) options operator, arrow: product(c(n)[j]-c(n)[j+1], j = 1 .. nops(c(n))-1) end proc: A := {}; for n to 200 do if pd(n) <> 0 then A := `union`(A, {n}) else end if end do: A; # most of the Maple program is due to W. Edwin Clark
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MATHEMATICA
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With[{nn = 18}, TakeWhile[#, # <= Floor[2^(2 + nn/Log2[nn])] &] &@ Union@ Apply[Join, #] &@ Table[Map[FromDigits[#, 2] &@ Flatten@ MapIndexed[ConstantArray[Boole@ OddQ@ #2, #1] &, #] &, Select[Map[Flatten[Map[# /. w_List :> If[First@ w == 1, Length@ w + 1, ConstantArray[1, Length@ w]] &, Split@ #] /. {a__, b_List, c__} :> {a, Most@ b, c}] &@ PadLeft[#, n - 1] &, IntegerDigits[Range[0, 2^n - 1], 2]], FreeQ[Differences@ #, 0] &]], {n, 2, nn}]] (* Michael De Vlieger, Jan 24 2018 *)
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CROSSREFS
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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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