A105529 -id:A105529 - cp's OEIS frontend

A128173 Numbers in ternary reflected Gray code order.

0, 1, 2, 5, 4, 3, 6, 7, 8, 17, 16, 15, 12, 13, 14, 11, 10, 9, 18, 19, 20, 23, 22, 21, 24, 25, 26, 53, 52, 51, 48, 49, 50, 47, 46, 45, 36, 37, 38, 41, 40, 39, 42, 43, 44, 35, 34, 33, 30, 31, 32, 29, 28, 27, 54, 55, 56, 59, 58, 57, 60, 61, 62, 71, 70, 69, 66, 67, 68, 65, 64, 63, 72

Offset: 0

Ralf Stephan, May 09 2007

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..19682
Donald E. Knuth, The Art of Computer Programming, Pre-Fascicle 2A, Draft of Section 7.2.1.1, subsection "Nonbinary Gray codes", page 18. At the bottom of page 19, ternary example g hat for all m_j=3 is the present sequence.
Z. Sunic, Tree morphisms, transducers and integer sequences, arXiv:math/0612080 [math.CO], 2006.
Index entries for sequences that are permutations of the natural numbers

Cf. A003188, A105529, A105530.

A128173 := proc(nmax) local K,tmp,n3,n,r,c,t,a ; n3 := 3 ; n := 1 ; K := linalg[matrix](n3,1,[[0],[1],[2]]) ; while n3 < nmax do n3 := n3*3 ; n := n+1 ; tmp := K ; K := linalg[extend](K,2*n3/3,1,0) ; K := linalg[copyinto](tmp,K,1+n3/3,1) ; K := linalg[copyinto](tmp,K,1+2*n3/3,1) ; for r from 1 to n3 do K[r,n] := floor((r-1)/(n3/3)) ; od ; for r from n3/3+1 to n3/2 do for c from 1 to n do t := K[r,c] ; K[r,c] := K[n3+1-r,c] ; K[n3+1-r,c] := t ; od ; od ; od ; a := [] ; for r from 1 to n3 do a := [op(a), add( K[r,c]*3^(c-1),c=1..n) ] ; od ; a ; end: A128173(30) ; # R. J. Mathar, Jun 17 2007

a[n_] := Module[{v, r, i}, v = IntegerDigits[n, 3]; r = 0; For[i = 1, i <= Length[v], i++, If[r == 1, v[[i]] = 2 - v[[i]]]; r = Mod[r + v[[i]], 2]]; FromDigits[v, 3]];
a /@ Range[0, 100] (* Jean-François Alcover, Jul 18 2020, after Kevin Ryde *)

a(n) = my(v=digits(n,3),r=Mod(0,2)); for(i=1,#v, if(r,v[i]=2-v[i]); r+=v[i]); fromdigits(v,3); \\ Kevin Ryde, May 21 2020

More terms from R. J. Mathar, Jun 17 2007

Offset changed to 0 by Alois P. Heinz, Feb 23 2018

A105530 Ternary modular Gray code for n.

Original entry on oeis.org

0, 1, 2, 5, 3, 4, 7, 8, 6, 15, 16, 17, 11, 9, 10, 13, 14, 12, 21, 22, 23, 26, 24, 25, 19, 20, 18, 45, 46, 47, 50, 48, 49, 52, 53, 51, 33, 34, 35, 29, 27, 28, 31, 32, 30, 39, 40, 41, 44, 42, 43

Offset: 0

Gary W. Adamson, Apr 11 2005

nonn

Ternary number n is converted into ternary Gray code a(n) by using the following algorithm: Leftmost term (i.e., digit) is leftmost Gray code term. Then going to the right, if next term b is greater than current term a, then (b - a) is the next Gray code term. (Gray code terms do not enter into the algorithmic operation). If next term b < a, then add 3 to b and perform [(3+b) - a] which becomes the next Gray code term. If b = a, the Gray code term = 0.
Interpreting any N-Ary code for n as N-Ary Gray code or vice versa results in a permutation of the natural numbers. Any N-Ary term can be converted to the N-Ary Gray code by using a generalization of the algorithmic rules such that if b < a, then add N to b and perform [(N + b) - a]. The other rules remain the same.
Inverse permutation of A105529.

Martin Cohn, Affine m-ary Gray Codes, Information and Control, volume 6, 1963, pages 70-78. (For the case m=3, U = P = identity matrix, v = 0 vector.)
Donald E. Knuth, The Art of Computer Programming, Pre-Fascicle 2A, Draft of Section 7.2.1.1. See subsection "Nonbinary Gray codes" page 18, and exercise 78 page 35 and answer page 54 (modular Gray g overline for the case all m_j=3).
Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (For p=3, stepping though switch position combinations by single changes.)
Index entries for sequences that are permutations of the natural numbers

Cf. A105529 (inverse), A128173 (ternary reflected), A003188 (binary), A098488 (decimal modular).

gray := proc(inp::integer,bas::integer) local resul, digs, convdigs, compl,d ; digs := [op(convert(inp,base,bas)),0] ; convdigs := [] ; for d from 1 to nops(digs)-1 do compl := op(d,digs)-op(d+1,digs) : if compl >= 0 then convdigs := [op(convdigs),compl] ; else convdigs := [op(convdigs),bas+compl] ; fi : od : convdigs := [op(convdigs),op(-1,digs)] : resul := 0 : for d from nops(convdigs) to 1 by -1 do resul := resul*bas + op(d,convdigs) : od : RETURN(resul) ; end: for n from 0 to 50 do printf("%a,",gray(n,3)) ; od : # R. J. Mathar, Mar 28 2006

a[n_] := Module[{v = IntegerDigits[n, 3]}, Do[v[[i]] = Mod[v[[i]] - v[[i-1]], 3], {i, Length[v], 2, -1}]; FromDigits[v, 3]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 26 2023, after Kevin Ryde *)

a(n) = my(v=digits(n,3)); forstep(i=#v,2,-1, v[i]=(v[i]-v[i-1])%3); fromdigits(v,3); \\ Kevin Ryde, May 23 2020

More terms from R. J. Mathar, Mar 28 2006

A370932 For any number n >= 0 with ternary expansion Sum_{i >= 0} t_i * 3^i, a(n) = Sum_{i >= 0} ((Sum_{j >= 0} (-1)^j * t_{i+j}) mod 3) * 3^i.

Original entry on oeis.org

0, 1, 2, 5, 3, 4, 7, 8, 6, 16, 17, 15, 9, 10, 11, 14, 12, 13, 23, 21, 22, 25, 26, 24, 18, 19, 20, 50, 48, 49, 52, 53, 51, 45, 46, 47, 27, 28, 29, 32, 30, 31, 34, 35, 33, 43, 44, 42, 36, 37, 38, 41, 39, 40, 70, 71, 69, 63, 64, 65, 68, 66, 67, 77, 75, 76, 79, 80

Offset: 0

Rémy Sigrist, Mar 06 2024

In other words, the k-th ternary digit of a(n) is congruent (modulo 3) to the alternate sum of the digits to the left of (and including) the k-th ternary digit of n.

This sequence is a permutation of the nonnegative integers with inverse A071770 that preserves the number of ternary digits (A081604) and the leading ternary digit (A122586).

			For n = 42: the ternary expansion of 42 is "1120"; also:
     + 1             = 1 (mod 3)
     - 1 + 1         = 0 (mod 3)
     + 1 - 1 + 2     = 2 (mod 3)
     - 1 + 1 - 2 + 0 = 1 (mod 3)
- so the ternary expansion of a(42) is "1021", and a(42) = 34.

Cf. A006068 (base-2 analog), A081604, A105529, A122586, A071770 (inverse).

a(n, base = 3) = { my (d = digits(n, base), s = 0); for (i = 1, #d, d[i] = (s = d[i]-s) % base;); fromdigits(d, base); }

from itertools import accumulate
from sympy.ntheory import digits
def A370932(n):
    t = accumulate(((-j if i&1 else j) for i, j in enumerate(digits(n,3)[1:])),func=lambda x,y: (x+y)%3)
    return int(''.join(str(-d%3 if i&1 else d) for i,d in enumerate(t)),3) # Chai Wah Wu, Mar 08 2024

A081604(a(n)) = A081604(n).

A122586(a(n)) = A122586(n).

A247891 a(n) is Gray-coded in base 4 into n.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 4, 10, 11, 8, 9, 15, 12, 13, 14, 21, 22, 23, 20, 26, 27, 24, 25, 31, 28, 29, 30, 16, 17, 18, 19, 42, 43, 40, 41, 47, 44, 45, 46, 32, 33, 34, 35, 37, 38, 39, 36, 63, 60, 61, 62, 48, 49, 50, 51, 53, 54, 55, 52, 58, 59, 56, 57, 85, 86, 87, 84, 90

Offset: 0

Alex Ratushnyak, Sep 26 2014

Permutation of nonnegative numbers.

Index entries for sequences that are permutations of the natural numbers

Cf. A006068 (base 2), A105529 (base 3), A226134 (base 10).

def basexor(a, b, base):
  result = 0
  digit = 1
  while a or b:
    da = a % base
    db = b % base
    a //= base
    b //= base
    sum = (da+db) % base
    result += sum * digit
    digit *= base
  return result
base = 4  #  2 => A006068, 3 => A105529, 10 => A226134
for n in range(129):
  a = n
  b = n//base
  while b:
    a = basexor(a,b, base)
    b //= base
  print(a, end=', ')

def xor4(x, y): return 4*xor4(x//4, y//4) + (x+y)%4 if x else y
def a(n): return xor4(n, a(n//4)) if n else 0 # David Radcliffe, Jun 22 2025

a(n) = n XOR4 [n/4] XOR4 [n/16] XOR4 [n/64] ... XOR4 [n/4^m] where m = [log(n)/log(4)], [x] is integer floor of x, and XOR4 is a base 4 analog of binary exclusive-OR operator.

In other words, a(n) = n + [n/4] + [n/16] + ... where addition is performed in base 4 without carries. - David Radcliffe, Jun 22 2025

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A128173 Numbers in ternary reflected Gray code order.

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A105530 Ternary modular Gray code for n.

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A370932 For any number n >= 0 with ternary expansion Sum_{i >= 0} t_i * 3^i, a(n) = Sum_{i >= 0} ((Sum_{j >= 0} (-1)^j * t_{i+j}) mod 3) * 3^i.

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A247891 a(n) is Gray-coded in base 4 into n.

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