A105530 Ternary modular Gray code for n.

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

Author

Gary W. Adamson, Apr 11 2005

Keywords

• nonn

Comments

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.

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Extensions

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