A026178 a(n) is the position of n in A026177.

1, 3, 4, 2, 7, 9, 10, 12, 13, 5, 16, 6, 19, 21, 22, 8, 25, 27, 28, 30, 31, 11, 34, 36, 37, 39, 40, 14, 43, 15, 46, 48, 49, 17, 52, 18, 55, 57, 58, 20, 61, 63, 64, 66, 67, 23, 70, 24, 73, 75, 76, 26, 79, 81, 82, 84, 85, 29, 88, 90, 91, 93, 94

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Inverse of A026177.

From Kevin Ryde, Feb 06 2020: (Start)

The formulas below are seen by considering values taken in the inverse A026177. Dekking's case II there is here a(2n) = n when lowest non-0 ternary digit LNZ(n) = A060236(n) = 2, which is LNZ(2n)=1. Otherwise case III is here a(2n+1) = 3n+1 and case IV is here a(2n) = 3n. These can be combined by a floor (the inverse of the ceiling in A026177).

For positive c with LNZ(c)=1, so c = (3j+1)*3^k = A026225, this sequence has a self-similarity in that taking the values which are multiples of c, and dividing them by c, gives the full sequence again. Using the inverse A026177, a(n)=c*m here is located at n = A026177(c*m) = c*A026177(m) - (floor(c/3) if m==1 (mod 3)) since c*m goes to the same bigger or smaller case in A026177 as m does. Then since floor(c/2) < c, values c*m are in the same order as all values m.

(End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• F. M. Dekking, Permutations of N generated by left-right filling algorithms, arXiv:2001.08915 [math.CO], 2020.

Crossrefs

Cf. A026177 (inverse).

Programs

Mathematica

Values[Take[#, LengthWhile[Differences@ Keys@ #, # == 1 &]]][[All, 1]] &@ KeySort@ PositionIndex@ Array[If[Mod[#/3^IntegerExponent[#, 3], 3] == 1, Ceiling[2 #/3], 2 #] &, 94] (* or *)
Block[{nn = 63, s}, s = Array[If[Mod[#/3^IntegerExponent[#, 3], 3] == 1, Ceiling[2 #/3], 2 #] &, Ceiling[3 nn/2]]; Array[FirstPosition[s, #][[1]] &, nn]] (* Michael De Vlieger, Feb 08 2020 *)

PARI

a(n) = if(n%2 || (n/3^valuation(n,3))%3==2, floor(3*n/2), n/2);

Formula

From Kevin Ryde, Feb 06 2020: (Start)

a(n) = n/2 if n even and A060236(n)=1, otherwise a(n) = floor(3n/2), where A060236(n) is the lowest non-0 ternary digit of n.

a(3n) = 3*a(n) + (1 if n odd).

(End)

Extensions

Edited by N. J. A. Sloane, Feb 05 2020