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A165317
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a(n) = the number of digits in the binary representation of n that each do not precede or follow a similarly valued digit.
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1
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1, 2, 0, 1, 3, 1, 0, 1, 2, 4, 2, 0, 2, 1, 0, 1, 2, 3, 1, 3, 5, 3, 2, 0, 1, 3, 1, 0, 2, 1, 0, 1, 2, 3, 1, 2, 4, 2, 1, 3, 4, 6, 4, 2, 4, 3, 2, 0, 1, 2, 0, 2, 4, 2, 1, 0, 1, 3, 1, 0, 2, 1, 0, 1, 2, 3, 1, 2, 4, 2, 1, 2, 3, 5, 3, 1, 3, 2, 1, 3, 4, 5, 3, 5, 7, 5, 4, 2, 3, 5, 3, 2, 4, 3, 2, 0, 1, 2, 0, 1, 3, 1, 0, 2, 3
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OFFSET
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1,2
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COMMENTS
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A165316(n) + a(n) = the number of digits in the binary representation of n.
Also number of parts equal to 1 in the composition having index n. For the definition of the index of a composition see A298644. For example, a(18) = 3 since the binary form of 18 is (1)00(1)(0) which has 3 runs of length 1 (each placed between parentheses). The command c(n) from the Maple program yields the composition having index n. - Emeric Deutsch, Jan 29 2018
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LINKS
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EXAMPLE
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184 in binary is 10111001. There are exactly three binary digits (the first and last 1's, and the leftmost 0) that are each not adjacent to a similar digit. So a(184) = 3.
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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: a := proc (n) local ct, j: ct := 0: for j to nops(c(n)) do if c(n)[j] = 1 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 105); # most of the Maple program is due to W. Edwin Clark. # Emeric Deutsch, Jan 29 2018
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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