A062756 Number of 1's in ternary (base-3) expansion of n.

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

Offset: 0

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 16 2001

Keywords

• nonn
• base

Comments

Fixed point of the morphism: 0 ->010; 1 ->121; 2 ->232; ...; n -> n(n+1)n, starting from a(0)=0. - Philippe Deléham, Oct 25 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.
• Michael Gilleland, Some Self-Similar Integer Sequences
• S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
• Kevin Ryde, Iterations of the Terdragon Curve, see index "dir".
• Robert Walker, Self Similar Sloth Canon Number Sequences

Crossrefs

Cf. A080846, A343785 (first differences).

Cf. A081606 (indices of !=0).

Indices of terms 0..6: A005823, A023692, A023693, A023694, A023695, A023696, A023697.

Numbers of: A077267 (0's), A081603 (2's), A160384 (1's+2's).

Other bases: A000120, A160381, A268643.

Programs

Haskell

a062756 0 = 0
a062756 n = a062756 n' + m `mod` 2 where (n',m) = divMod n 3
-- Reinhard Zumkeller, Feb 21 2013

Mathematica

Table[Count[IntegerDigits[i, 3], 1], {i, 0, 200}]
Nest[Join[#, # + 1, #] &, {0}, 5] (* IWABUCHI Yu(u)ki, Sep 08 2012 *)

PARI

a(n)=if(n<1,0,a(n\3)+(n%3)%2) \\ Paul D. Hanna, Feb 24 2006

PARI

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

Python

from sympy.ntheory import digits
def A062756(n): return digits(n,3)[1:].count(1) # Chai Wah Wu, Dec 23 2022

Formula

a(0) = 0, a(3n) = a(n), a(3n+1) = a(n)+1, a(3n+2) = a(n). - Vladeta Jovovic, Jul 18 2001

G.f.: (Sum_{k>=0} x^(3^k)/(1+x^(3^k)+x^(2*3^k)))/(1-x). In general, the generating function for the number of digits equal to d in the base b representation of n (0 < d < b) is (Sum_{k>=0} x^(d*b^k)/(Sum_{i=0..b-1} x^(i*b^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005 [For d=0, use the above formula with d=b: (Sum_{k>=0} x^(b^(k+1))/(Sum_{i=0..b-1} x^(i*b^k)))/(1-x), adding 1 if you consider the representation of 0 to have one zero digit.]

a(n) = a(floor(n/3)) + (n mod 3) mod 2. - Paul D. Hanna, Feb 24 2006

Extensions

More terms from Vladeta Jovovic, Jul 18 2001