A081603 Number of 2's in ternary representation of n.

0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 3, 4, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2

Offset: 0

Author

Reinhard Zumkeller, Mar 23 2003

Keywords

• nonn
• base

Comments

Fixed point of the morphism: 0 ->001; 1 ->112; 2 ->223; 3 ->334, etc., starting from a(0)=0. - Philippe Deléham, Oct 26 2011

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.
• Eric Weisstein's World of Mathematics, Ternary.

Crossrefs

Cf. A007089, A074940, A005836, A081610, A081611, A117592 (2^a(n)).

Programs

Haskell

a081603 0 = 0
a081603 n = a081603 n' + m `div` 2 where (n',m) = divMod n 3
-- Reinhard Zumkeller, Feb 21 2013

Maple

A081603 := proc(n)
    local a,d ;
    a := 0 ;
    for d in convert(n,base,3) do
        if d= 2 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jul 12 2016

Mathematica

Table[Count[IntegerDigits[n,3],2],{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Jul 25 2009 *)
Nest[ Flatten[# /. a_Integer -> {a, a, a + 1}] &, {0}, 5] (* Robert G. Wilson v, May 20 2014 *)
DigitCount[Range[0,120],3,2] (* Harvey P. Dale, Jul 10 2019 *)

PARI

a(n)=hammingweight(digits(n,3)\2); \\ Ruud H.G. van Tol, Dec 10 2023

Python

from gmpy2 import digits
def A081603(n): return digits(n,3).count('2') # Chai Wah Wu, Dec 05 2024

Formula

a(n) = floor(n/2) if n < 3, otherwise a(floor(n/3)) + floor((n mod 3)/2).

A077267(n) + A062756(n) + a(n) = A081604(n);

a(n) = (A053735(n) - A062756(n))/2.