A024629 n written in fractional base 3/2.

0, 1, 2, 20, 21, 22, 210, 211, 212, 2100, 2101, 2102, 2120, 2121, 2122, 21010, 21011, 21012, 21200, 21201, 21202, 21220, 21221, 21222, 210110, 210111, 210112, 212000, 212001, 212002, 212020, 212021, 212022, 212210, 212211, 212212, 2101100, 2101101

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• base

Comments

A246435(n) = (number of digits in a(n)) = A055642(a(n)). - Reinhard Zumkeller, Sep 05 2014

The number of positive even n such that a(n) has k+1 digits is A005428(k). - Glen Whitney, Jul 09 2017

The position of the rightmost "2" digit in a(3k), k >= 1, appears to be A087088(k). - Peter Munn, Jun 24 2020 [updated Peter Munn, Jul 14 2020 for new A087088 offset]

Examples

			Representations of the first few numbers are:
   0 =         0
   1 =         1
   2 =         2
   3 =       2 0
   4 =       2 1
   5 =       2 2
   6 =     2 1 0
   7 =     2 1 1
   8 =     2 1 2
   9 =   2 1 0 0
  10 =   2 1 0 1
  11 =   2 1 0 2
  12 =   2 1 2 0
  13 =   2 1 2 1
  14 =   2 1 2 2
  15 = 2 1 0 1 0
[extended and reformatted by _Peter Munn_, Jun 27 2020]
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, and Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019.
• B. Chen, R. Chen, J. Guo, S. Lee et al, On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
• Michel Dekking, The Thue-Morse sequence in base 3/2, arXiv:2301.13563 [math.CO], 2023. See also J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
• Tanya Khovanova and Kevin Wu, Base 3/2 and Greedily Partitioned Sequences, arXiv:2007.09705 [math.NT], 2020.
• J. S. Tanton, A collection of research problems. [archived version]

Crossrefs

Cf. A081848, A087088, A246435 (string lengths), A244040 (digit sums).

Programs

Haskell

a024629 0 = 0
a024629 n = 10 * a024629 (2 * n') + t where (n', t) = divMod n 3
-- Reinhard Zumkeller, Sep 05 2014

Maple

a:= proc(n) `if`(n<1, 0, irem(n, 3, 'q')+a(2*q)*10) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jun 19 2018

Mathematica

a[ n_] := If[ n < 1, 0, a[ Quotient[n, 3] 2] 10 + Mod[ n, 3]]; (* Michael Somos, Jun 18 2014 *)

PARI

{a(n) = if( n<1, 0, a(n\3 * 2) * 10 + n%3)}; /* Michael Somos, Jun 18 2014 */

SageMath

def basepqExpansion(p,q,n):
    L, i = [n], 1
    while L[i-1] >= p:
        x=L[i-1]
        L[i-1]=x.mod(p)
        L.append(q*(x//p))
        i+=1
    L.reverse()
    return Integer(''.join(str(x) for x in L))
[basepqExpansion(3,2,n) for n in [0..40]] # Tom Edgar, Hailey R. Olafson, and James Van Alstine, Jun 17 2014; modified and corrected by G. C. Greubel, Aug 20 2019

Formula

To represent a number in base b, if a digit is >= b, subtract b and carry 1. In fractional base a/b, subtract a and carry b.

a(0)=0, a(3n+r) = 10*a(2n)+r for n >= 0 and r = 0, 1, 2. - Jianing Song, Oct 15 2022

Extensions

Tanton link corrected by Charles R Greathouse IV, Oct 20 2008