A160384 -id:A160384 - cp's OEIS frontend

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

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

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

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

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.

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

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

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

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

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

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

More terms from Vladeta Jovovic, Jul 18 2001

A343601 For any positive number n, the ternary representation of a(n) is obtained by right-rotating the ternary representation of n until a nonzero digit appears again as the leftmost digit; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 12, 21, 10, 13, 22, 19, 14, 23, 18, 15, 24, 11, 16, 25, 20, 17, 26, 27, 36, 63, 30, 37, 64, 57, 38, 65, 28, 39, 66, 31, 40, 67, 58, 41, 68, 55, 42, 69, 32, 43, 70, 59, 44, 71, 54, 45, 72, 33, 46, 73, 60, 47, 74, 29, 48, 75, 34, 49

Offset: 0

Rémy Sigrist, Apr 21 2021

This sequence is a permutation of the nonnegative integers with inverse A343600.

			The first terms, in base 10 and in base 3, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2       2          2
   3     3      10         10
   4     4      11         11
   5     7      12         21
   6     6      20         20
   7     5      21         12
   8     8      22         22
   9     9     100        100
  10    12     101        110
  11    21     102        210
  12    10     110        101
  13    13     111        111
  14    22     112        211

Cf. A053735, A081604, A139706 (binary variant), A160384, A343600 (inverse).

a(n, base=3) = { my (d=digits(n, base)); forstep (k=#d, 2, -1, if (d[k], return (fromdigits(concat(d[k..#d], d[1..k-1]), base)))); n }

A053735(a(n)) = A053735(n).
A081604(a(n)) = A081604(n).
a^k(n) = n for k = A160384(n) (where a^k denotes the k-th iterate of a).

A300956 a(0) = 0, a(1) = 2, a(2) = 1, and for any n > 2 with ternary representation n = Sum_{i=0..k} t_i * 3^i, a(n) = Sum_{i=0..k} a(t_i) * 3^a(i).

Original entry on oeis.org

0, 2, 1, 18, 20, 19, 9, 11, 10, 6, 8, 7, 24, 26, 25, 15, 17, 16, 3, 5, 4, 21, 23, 22, 12, 14, 13, 774840978, 774840980, 774840979, 774840996, 774840998, 774840997, 774840987, 774840989, 774840988, 774840984, 774840986, 774840985, 774841002, 774841004

Offset: 0

Rémy Sigrist, Mar 17 2018

This sequence is a self-inverse permutation of the natural numbers.
This sequence has connections with A300955.
This sequence has infinitely many fixed points (A300958); for any k >= 0, at least one of k or 3^k + 2 * 3^a(k) is a fixed point.

Cf. A160384, A300955, A300958 (fixed points).

a(n) = my (t=Vecrev(digits(n,3))); sum(i=0, #t-1, if (t[i+1]==1, 2, t[i+1]==2, 1, 0) * 3 ^ a(i))

A160384(a(n)) = A160384(n).

a(a(n)) = n.

A343600 For any positive number n, the ternary representation of a(n) is obtained by left-rotating the ternary representation of n until a nonzero digit appears again as the leftmost digit; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 12, 21, 10, 13, 16, 19, 22, 25, 18, 15, 24, 11, 14, 17, 20, 23, 26, 27, 36, 63, 30, 39, 48, 57, 66, 75, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 54, 45, 72, 33, 42, 51, 60, 69, 78, 29, 32, 35, 38, 41

Offset: 0

Rémy Sigrist, Apr 21 2021

This sequence is a permutation of the nonnegative integers with inverse A343601.

			The first terms, in base 10 and in base 3, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2       2          2
   3     3      10         10
   4     4      11         11
   5     7      12         21
   6     6      20         20
   7     5      21         12
   8     8      22         22
   9     9     100        100
  10    12     101        110
  11    21     102        210
  12    10     110        101
  13    13     111        111
  14    16     112        121

Cf. A053735, A081604, A139708 (binary variant), A160384, A343601 (inverse).

a(n, base=3) = { my (d=digits(n, base)); for (k=2, #d, if (d[k], return (fromdigits(concat(d[k..#d], d[1..k-1]), base)))); n }

A053735(a(n)) = A053735(n).
A081604(a(n)) = A081604(n).
a^k(n) = n for k = A160384(n) (where a^k denotes the k-th iterate of a).

A333773 Replace 2's with (-1)'s in ternary representation of n and sum nonzero terms with alternating signs.

Original entry on oeis.org

0, 1, -1, 3, 2, 4, -3, -4, -2, 9, 8, 10, 6, 7, 5, 12, 13, 11, -9, -10, -8, -12, -11, -13, -6, -5, -7, 27, 26, 28, 24, 25, 23, 30, 31, 29, 18, 19, 17, 21, 20, 22, 15, 14, 16, 36, 37, 35, 39, 38, 40, 33, 32, 34, -27, -28, -26, -30, -29, -31, -24, -23, -25, -36

Offset: 0

Rémy Sigrist, Apr 05 2020

This sequence is a variant of A117966, and shares features with A065620.

Every integer appears exactly once in this sequence.

			For n = 97:
- 97 = 3^4 + 3^2 + 2*3^1 + 3^0,
- hence a(97) = 3^4 - 3^2 + (-1)*3^1 - 3^0 = 68.

Rémy Sigrist, Table of n, a(n) for n = 0..6561

Cf. A004488, A024023, A065620, A117966, A132141, A157671, A160384, A333780.

a(n) = { my (v=0, t=Vecrev(digits(n,3))); for (k=1, #t, if (t[k]==1, v=+3^(k-1)-v, t[k]==2, v=-3^(k-1)-v)); v }

a(3*n)   = 3*a(n).
a(3*n+1) = 3*a(n) + (-1)^A160384(n).
a(3*n+2) = 3*a(n) - (-1)^A160384(n).
Sum_{k=0..n} a(k) >= 0 with equality iff n belongs to A024023.
a(n) > 0 iff n belongs to A132141.
a(n) < 0 iff n belongs to A157671.
a(A004488(n)) = -a(n).

A206567 S(m,n) = (number of nonzero terms common to the base 3 expansions of m and n), a symmetric matrix read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1

Offset: 1

Clark Kimberling, Feb 09 2012

Every nonnegative integer occurs infinitely many times in the matrix.

			Northwest corner:
1 0 0 1 0 0 1 0 0 1 0 0 1
0 1 0 0 1 0 0 1 0 0 1 0 0
0 0 1 1 1 0 0 0 0 0 0 1 1
1 0 1 2 1 0 1 0 0 1 0 1 2
0 1 1 1 2 0 0 1 0 0 1 1 1
0 0 0 0 0 1 1 1 0 0 0 0 0
1 0 0 1 0 1 2 1 0 1 0 0 1
0 1 0 0 1 1 1 2 0 0 1 0 0
0 0 0 0 0 0 0 0 1 1 1 1 1
1 0 0 1 0 0 1 0 1 2 1 1 2
0 1 0 0 1 0 0 1 1 1 2 1 1
0 0 1 1 1 0 0 0 1 1 1 2 2
1 0 1 2 1 0 1 0 1 2 1 2 3
4 = 3 + 1 and 13 = 3^2 + 3 + 1, so S(13,4)=2.

Robert Israel, Table of n, a(n) for n = 1..10011 (antidiagonals 1 to 141, flattened)

Cf. A160384, A206479 (similar in base 2).

S:= proc(m,n) local M,N;
  M:= convert(m,base,3);
  N:= convert(n,base,3);
  convert(zip((s,t) -> `if`(s=t and s <> 0, 1, 0),M,N),`+`);
end proc:
seq(seq(S(k,n-k+1),k=1..n),n=1..30); # Robert Israel, Mar 19 2018

d[n_] := IntegerDigits[n, 3];
t[n_] := Reverse[Array[d, 100][[n]]]
s[n_, k_] := Position[t[n], k]
t[m_, n_] := Sum[Length[Intersection[s[m, k], s[n, k]]], {k, 1, 2}]
TableForm[Table[t[m, n], {m, 1, 24},
  {n, 1, 24}]]  (* A206567 as a matrix *)
Flatten[Table[t[i, n + 1 - i], {n, 1, 24},
  {i, 1, n}]]   (* A206567 as a sequence *)

d(n) = Vecrev(digits(n, 3));
T(n, k) = {my(dn = d(n), dk = d(k), nb = min(#dn, #dk)); sum(i=1, nb, dn[i] && (dn[i] == dk[i]));} \\ Michel Marcus, Mar 19 2018

Diagonal entries S(n,n) = A160384(n) since all nonzero digits match. - Robert Israel, Mar 18 2018

Edited by Robert Israel, Mar 19 2018

A234538 (Number of positive digits of n written in base 3) modulo 3.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 0, 0, 2, 0, 0, 1, 2, 2, 2, 0, 0, 2, 0, 0, 1, 2, 2, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 2, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 2, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1

Offset: 0

Vladimir Shevelev, Jan 13 2014

Since A000120 is the number of positive digits of n written in binary, this sequence is a formal ternary analog of the Thue-Morse sequence A010060. However, one cannot name it a "ternary version of A010060" like the known versions A053838, A071858, A036577-A036586, since it is not "cubefree"; i.e., it contains the same 3 consecutive terms, and there is not a known morphism for which it is a fixed point.

Peter J. C. Moses, Table of n, a(n) for n = 0..999

Cf. A000120, A010060, A036577-A036586, A053838, A071858, A160384.

a[n_] := Mod[Plus @@ DigitCount[n, 3, {1, 2}], 3]; Array[a, 100, 0] (* Amiram Eldar, Jul 24 2023 *)

a(n)=my(d=digits(n, 3)); sum(i=1, #d, !d[i])%3 \\ Charles R Greathouse IV, Jan 13 2014

A160384(n) == a(n) (mod 3).

A276134 a(5n) = a(n), a(5n+1) = a(5n+2) = a(5n+3) = a(5n+4) = a(n) + 1, a(0) = 0.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 21 2016

Number of nonzero digits in the base 5 representation of n.
Fixed point of the mapping 0 -> 01111, 1 -> 12222, 2 -> 23333, ...
Self-similar or fractal sequence (underlining every fifth term, reproduce the original sequence).

			The evolution starting with 0 is: 0 -> 01111 -> 0111112222122221222212222 -> ...
...
a(0) = 0;
a(1) = a(5*0+1) = a(0) + 1 = 1;
a(2) = a(5*0+2) = a(0) + 1 = 1;
a(3) = a(5*0+3) = a(0) + 1 = 1;
a(4) = a(5*0+4) = a(0) + 1 = 1;
a(5) = a(5*1+0) = a(1) = 1;
a(6) = a(5*1+1) = a(1) + 1 = 2, etc.
...
Also a(10) = 1, because 10 (base 10) = 20 (base 5) and 20 has 1 nonzero digit.

Robert Israel, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Illustration (mapping 0 -> 01111, 1 -> 12222, 2 -> 23333, ...)

Cf. A000034, A000120, A093178, A160384, A160385.

f:= n -> nops(subs(0=NULL,convert(n,base,5))):
map(f, [$0..100]); # Robert Israel, Sep 07 2016

Join[{0}, Table[IntegerLength[n, 5] - DigitCount[n, 5, 0], {n, 120}]]

a(5^k) = 1.
a(5^k-1) = k.
a(5^k-m) = k, k>0, m = 2,3,4.
a(5^k+m) = 2, k>0, m = 1,2,3,4.
a(5^k-a(5^k)) = k.
a(5^k+(-1)^k) = (k + (-1)^k*(k - 1) + 3)/2.
a(5^k+(-1)^k-1) = A093178(k).
a(5^k+(-1)^k+1) = A000034(k+1), k>0.
G.f. g(x) satisfies g(x) = (1+x+x^2+x^3+x^4)*g(x^5) + (x+x^2+x^3+x^4)/(1-x^5). - Robert Israel, Sep 07 2016

A366344 Irregular triangle T(n, k), n >= 0, k = 1 or 2, read by rows; the n-th row contains two coprime positive integers whose prime factorizations are encoded in the ternary expansion of n (see Comments section for precise definition).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 3, 2, 1, 3, 2, 3, 1, 4, 5, 1, 6, 1, 5, 2, 9, 1, 8, 1, 9, 2, 5, 3, 10, 3, 3, 4, 1, 5, 2, 5, 1, 6, 3, 5, 4, 3, 3, 10, 1, 9, 2, 9, 1, 8, 7, 1, 10, 1, 7, 2, 15, 1, 12, 1, 15, 2, 7, 3, 14, 3, 5, 4, 25, 1, 18, 1, 25, 2, 27, 1, 16, 1, 27, 2

Offset: 0

Rémy Sigrist, Oct 07 2023

The encoding used here is related to that used for the Doudna sequence (A005940):
- for any pair (u, v) of coprime positive integers, the ternary expansion of the unique n >= 0 such that T(n, 1) = u and T(n, 2) = v is built as follows (from right to left):
   - for m = 1, 2, ..., let p be the m-th prime number,
     - if p neither divides u nor v then we add a 0,
     - if p divides u with multiplicity e then we add a run of e 1's,
     - if p divides v with multiplicity e then we add a run of e 2's,
     - we also insert an extra 0 between pairs of runs of 1's not separated by 2's and between pairs of runs of 2's not separated by 1's.
This encoding can be applied to any fixed base b >= 2 and will yield a bijection from the nonnegative integers to the set of tuples of b-1 pairwise coprime positive integers.
The case b = 2 corresponds (up to the offset) to the Doudna sequence (A005940).
The sequence n -> T(n, 1) / T(n, 2) runs through all the reduced positive rationals exactly once.

			Triangle T(n, k) begins (alongside the ternary expansion of n):
  n   n-th row  ter(n)
  --  --------  ------
   0  [1, 1]         0
   1  [2, 1]         1
   2  [1, 2]         2
   3  [3, 1]        10
   4  [4, 1]        11
   5  [3, 2]        12
   6  [1, 3]        20
   7  [2, 3]        21
   8  [1, 4]        22
   9  [5, 1]       100
  10  [6, 1]       101
  11  [5, 2]       102
  12  [9, 1]       110
  13  [8, 1]       111
  14  [9, 2]       112
  15  [5, 3]       120
  16  [10, 3]      121
  17  [3, 4]       122

Cf. A000040, A001222, A003961, A004488, A005823, A005836, A005940, A062756, A081603, A160384.

row(n, b = 3) = { my (r = vector(b-1, d, 1), g = 0, t = 0); while (n, my (d = n % b); n \= b; g++; if (d, my (e = 1); while (n % b == d, e++; n \= b;); if (t==d, g--, t = d); r[d] *= prime(g)^e;);); return (r); }

T(n, 1) = 1 iff n belongs to A005823.
T(n, 2) = 1 iff n belongs to A005836.
T(A005836(n), 1) = A005940(n+1).
T(A005823(n), 2) = A005940(n+1).
A001222(T(n, 1)) = A062756(n).
A001222(T(n, 2)) = A081603(n).
A001222(T(n, 1) * T(n, 2)) = A160384(n).
T(A004488(n), 1) = T(n, 2).
T(A004488(n), 2) = T(n, 1).
T((3^e - 1)/2, 1) = 2^e for any e >= 0.
T(3^e - 1, 2) = 2^e for any e >= 0.
T(3^e, 1) = A000040(e + 1) for any e >= 0.
T(2 * 3^e, 2) = A000040(e + 1) for any e >= 0.
T(3*n, k) = A003961(T(n, k)).

cp's OEIS Frontend

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

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A343601 For any positive number n, the ternary representation of a(n) is obtained by right-rotating the ternary representation of n until a nonzero digit appears again as the leftmost digit; a(0) = 0.

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A300956 a(0) = 0, a(1) = 2, a(2) = 1, and for any n > 2 with ternary representation n = Sum_{i=0..k} t_i * 3^i, a(n) = Sum_{i=0..k} a(t_i) * 3^a(i).

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A343600 For any positive number n, the ternary representation of a(n) is obtained by left-rotating the ternary representation of n until a nonzero digit appears again as the leftmost digit; a(0) = 0.

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A333773 Replace 2's with (-1)'s in ternary representation of n and sum nonzero terms with alternating signs.

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A206567 S(m,n) = (number of nonzero terms common to the base 3 expansions of m and n), a symmetric matrix read by antidiagonals.

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A234538 (Number of positive digits of n written in base 3) modulo 3.

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