A278985 List of words of length n over an alphabet of size 3 that are in standard order.

1, 11, 12, 111, 112, 121, 122, 123, 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 11111, 11112, 11121, 11122, 11123, 11211, 11212, 11213, 11221, 11222, 11223, 11231, 11232, 11233, 12111, 12112, 12113, 12121, 12122

Offset

1,2

Comments

We study words made of letters from an alphabet of size b, where b >= 1. We assume the letters are labeled {1,2,3,...,b}. There are b^n possible words of length n.

We say that a word is in "standard order" if it has the property that whenever a letter i appears, the letter i-1 has already appeared in the word. This implies that all words begin with the letter 1.

These are the words described in row b=3 of the array in A278984.

A007051(n-1) gives the number of n-digit terms in this sequence. - Rémy Sigrist, Dec 18 2016

Links

Rémy Sigrist and N. J. A. Sloane, Table of n, a(n) for n = 1..14767 [Terms 1 through 185 by N. J. A. Sloane]

Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

Maple

b:= proc(n) option remember; `if`(n=1, [[1]], map(x->
      seq([x[], i], i=1..min(3, max(x[])+1)), b(n-1)))
    end:
T:= n-> map(x-> parse(cat(x[])), b(n))[]:
seq(T(n), n=1..5);  # Alois P. Heinz, Jan 02 2022

Mathematica

Table[FromDigits /@ Select[Tuples[Range@ 3, n], And[Times @@ Boole@ MapIndexed[#1 <= First@ #2 &, #] > 0, Max@ Differences@ # <= 1] &], {n, 5}] // Flatten (* Michael De Vlieger, Dec 18 2016 *)

Prog

(PARI) gen(n, len, mx) = if (len==0, print1 (n ", "), for (d=1, min(mx+1, 3), gen(10*n + d, len-1, max(mx, d))))
for (len=1, 5, gen(0, len, 0)) \\ Rémy Sigrist, Dec 18 2016

Crossrefs

Cf. A278984, A007051.

Similar to but different from A071159.

Sequence in context: A239463 A153070 A193023 * A071159 A231873 A096299

Adjacent sequences: A278982 A278983 A278984 * A278986 A278987 A278988

Keyword

nonn,base

Author

N. J. A. Sloane, Dec 05 2016

Status

approved