A162853 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.

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

Offset

0,2

Comments

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

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = A266150(A266151(n)) = A266151(A266150(n)) for any n > 0. - Rémy Sigrist, Oct 09 2018

Example

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.

Maple

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

Mathematica

Table[FromDigits[Flatten[If[OddQ[Length[#]], Join[{First[#]}, #], Drop[#, 1]]& /@Split[ IntegerDigits[ n, 2]]], 2], {n, 70}] (* Harvey P. Dale, Jun 20 2011 *)

Prog

(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

Crossrefs

Cf. A014681, A266150, A266151.

Sequence in context: A181405 A354568 A072117 * A162854 A342787 A110121

Adjacent sequences: A162850 A162851 A162852 * A162854 A162855 A162856

Keyword

base,nonn

Author

Leroy Quet, Jul 14 2009

Extensions

Extended beyond a(13) by R. J. Mathar, Aug 01 2009

a(0) added by Rémy Sigrist, Oct 09 2018

Status

approved