A183209 Tree generated by floor(3n/2): a(1) = 1, a(2n) = (3*a(n))-1, a(2n+1) = floor((3*a(n+1))/2).

1, 2, 3, 5, 4, 8, 7, 14, 6, 11, 12, 23, 10, 20, 21, 41, 9, 17, 16, 32, 18, 35, 34, 68, 15, 29, 30, 59, 31, 62, 61, 122, 13, 26, 25, 50, 24, 47, 48, 95, 27, 53, 52, 104, 51, 101, 102, 203, 22, 44, 43, 86, 45, 89, 88, 176, 46, 92, 93, 185, 91, 182, 183, 365, 19, 38, 39, 77, 37

Offset: 1

Author

Clark Kimberling, Dec 30 2010

Keywords

• nonn
• tabf
• look

Comments

A permutation of the positive integers. See the comment at A183079. Leftmost branch of tree is essentially A061418. Rightmost: A007051.

Examples

			First levels of the tree:
                      1
                      2
            3                   5
          4   8               7   14
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192; the first 14 levels of the tree, flattened
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A183079, A007051, A016789, A183207, A183208, A032766, A191450.

Similar permutations: A048673, A254103.

Inverse permutation: A259431.

Programs

Haskell

import Data.List (transpose)
a183209 n k = a183209_tabf !! (n-1) !! (k-1)
a183209_row n = a183209_tabf !! (n-1)
a183209_tabf = [1] : iterate (\xs -> concat $
   transpose [map a032766 xs, map (a016789 . subtract 1) xs]) [2]
a183209_list = concat a183209_tabf
-- Reinhard Zumkeller, Jun 27 2015

Maple

f:= proc(n) option remember;
  if n::even then 3*procname(n/2)-1
  else floor(3*procname((n+1)/2)/2)
  fi
end proc:
f(1):= 1:
seq(f(n), n=1..100); # Robert Israel, Jan 26 2015

Mathematica

a[1]=1; a[n_] := a[n] = If[EvenQ[n], 3a[n/2]-1, Floor[3a[(n+1)/2]/2] ]; Array[a, 100] (* Jean-François Alcover, Feb 02 2018 *)

Python

def a(n):
    if n==1: return 1
    if n%2==0: return 3*a(n//2) - 1
    else: return (3*a((n - 1)//2 + 1))//2
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

Formula

Let L(n)=floor(3n/2).

Let U(n)=3n-1. U is the complement of L.

The tree-array T(n,k) is then given by rows:

T(0,0)=1; T(1,0)=2;

T(n,2j)=L(T(n-1),j);

T(n,2j+1)=U(T(n-1),j);

for j=0,1,...,2^(n-1)-1, n>=2.

From Antti Karttunen, Jan 26 2015: (Start)

a(1) = 1, a(2n) = (3*a(n))-1, a(2n+1) = A032766(a(n+1)) = floor((3*a(n+1))/2).

Other identities:

a(2^n) = A007051(n) for all n >= 0. [A property shared with A048673 and A254103.]

(End)

Extensions

Formula to the name-field added by Antti Karttunen, Jan 26 2015