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Number of 1's in ternary (base-3) expansion of n.
56

%I #63 Dec 10 2023 11:10:27

%S 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,

%T 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,

%U 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

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

%C 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

%H Reinhard Zumkeller, <a href="/A062756/b062756.txt">Table of n, a(n) for n = 0..10000</a>

%H F. T. Adams-Watters and F. Ruskey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ruskey2/ruskey14.html">Generating Functions for the Digital Sum and Other Digit Counting Sequences</a>, JIS 12 (2009) 09.5.6.

%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>

%H S. Northshield, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Northshield/north4.html">An Analogue of Stern's Sequence for Z[sqrt(2)]</a>, Journal of Integer Sequences, 18 (2015), #15.11.6.

%H Kevin Ryde, <a href="http://user42.tuxfamily.org/terdragon/index.html">Iterations of the Terdragon Curve</a>, see index "dir".

%H Robert Walker, <a href="http://robertinventor.com/ftswiki/Self_Similar_Sloth_Canon_Number_Sequences">Self Similar Sloth Canon Number Sequences</a>

%F a(0) = 0, a(3n) = a(n), a(3n+1) = a(n)+1, a(3n+2) = a(n). - _Vladeta Jovovic_, Jul 18 2001

%F 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.]

%F a(n) = a(floor(n/3)) + (n mod 3) mod 2. - _Paul D. Hanna_, Feb 24 2006

%t Table[Count[IntegerDigits[i, 3], 1], {i, 0, 200}]

%t Nest[Join[#, # + 1, #] &, {0}, 5] (* _IWABUCHI Yu(u)ki_, Sep 08 2012 *)

%o (PARI) a(n)=if(n<1,0,a(n\3)+(n%3)%2) \\ _Paul D. Hanna_, Feb 24 2006

%o (PARI) a(n)=hammingweight(digits(n,3)%2); \\ _Ruud H.G. van Tol_, Dec 10 2023

%o (Haskell)

%o a062756 0 = 0

%o a062756 n = a062756 n' + m `mod` 2 where (n',m) = divMod n 3

%o -- _Reinhard Zumkeller_, Feb 21 2013

%o (Python)

%o from sympy.ntheory import digits

%o def A062756(n): return digits(n,3)[1:].count(1) # _Chai Wah Wu_, Dec 23 2022

%Y Cf. A080846, A343785 (first differences).

%Y Cf. A081606 (indices of !=0).

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

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

%Y Other bases: A000120, A160381, A268643.

%K nonn,base

%O 0,5

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

%E More terms from _Vladeta Jovovic_, Jul 18 2001