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A162853
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Take the binary representation of n. Reduce by one digit every run (completely of either 0's or 1's) of an even number of digits. Increase by one digit every run of an odd number of digits in the binary representation of n (where this added digit has the same value that makes up the rest of the run's digits). a(n) = the decimal equivalent of the result.
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4
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0, 3, 12, 1, 6, 51, 4, 15, 48, 27, 204, 25, 2, 19, 60, 7, 24, 195, 108, 13, 102, 819, 100, 207, 16, 11, 76, 9, 30, 243, 28, 63, 192, 99, 780, 97, 54, 435, 52, 111, 816, 411, 3276, 409, 50, 403, 828, 103, 8, 67, 44, 5, 38, 307, 36, 79, 240, 123, 972, 121, 14, 115, 252, 31, 96
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OFFSET
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0,2
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COMMENTS
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This is a self-inverse permutation of the nonnegative integers.
Clarification: The consecutive "runs" (mentioned in the definition) alternate between those completely of 1's and those completely of 0's.
In the binary representation of n, replace each run of length r by a run of length A014681(r). - Rémy Sigrist, Oct 09 2018
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LINKS
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FORMULA
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EXAMPLE
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152 in binary is 10011000. There is a run of one 1, followed by a run of two 0's, followed by a run of two 1's, followed by a run of three 0's. We reduce the two runs of two digits each to one digit; and we add a digit (a 1) to the first run of one 1, and a digit (a 0) to the last run of three 0's, to get 11010000. So a(152) is the decimal equivalent of this, which is 208.
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MAPLE
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rerun := proc(L) if nops(L) mod 2 = 0 then subsop(1=NULL, L) ; else [op(L), op(1, L)] ; fi; end: Lton := proc(L) local i; add( op(i, L)*2^(i-1), i=1..nops(L)) ; end: A162853 := proc(n) local strt, en, L, dgs, i; strt := 1; en := -1; L := [] ; dgs := convert(n, base, 2) ; for i from 2 to nops(dgs) do if op(i, dgs) <> op(i-1, dgs) then en := i-1 ; L := [op(L), op(rerun([op(strt..en, dgs)])) ] ; strt := i; fi; od: en := nops(dgs) ; L := [op(L), op(rerun([op(strt..en, dgs)])) ] ; Lton(L) ; end: seq(A162853(n), n=1..100) ; # R. J. Mathar, Aug 01 2009
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MATHEMATICA
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Table[FromDigits[Flatten[If[OddQ[Length[#]], Join[{First[#]}, #], Drop[#, 1]]& /@Split[ IntegerDigits[ n, 2]]], 2], {n, 70}] (* Harvey P. Dale, Jun 20 2011 *)
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PROG
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(PARI) a(n) = if (n==0, 0, my (b=n%2, r=valuation(n+b, 2), rr=if (r%2, r+1, r-1)); (a(n\2^r)+b)*2^rr-b) \\ Rémy Sigrist, Oct 09 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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