A319154 a(n) is the smallest nonnegative integer not yet in the sequence that starts with the ending digit of a(n-1); a(1)=0; initial zeros are dropped.

0, 1, 10, 2, 20, 3, 30, 4, 40, 5, 50, 6, 60, 7, 70, 8, 80, 9, 90, 11, 12, 21, 13, 31, 14, 41, 15, 51, 16, 61, 17, 71, 18, 81, 19, 91, 100, 22, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 92, 200, 33, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 93, 300, 44, 45, 54

Offset

1,3

Comments

Theorem: Every nonnegative number appears.

Proof: (Sketched by Enrique Navarrete, Sep 25 2018; completed by N. J. A. Sloane, Oct 27 2018)

(i) Sequence is infinite (dG, G=giant number, is always available)

(ii) As usual for these "lexicographically earliest distinct term sequences", for any k, there is a threshold n_k such that for all n > n_k, a(n) > k.

(iii) Some final digit (d, say) appears infinitely often. (Otherwise sequence would be finite.) If d=0, go to step (vi).

(iv) All numbers beginning with d appear (If dm were missing, find xd in sequence which is > dm and also > n_{dm}. Then term after xd would be dm, contradiction.)

(v) In particular, all numbers dm0 appear.

(vi) After a number ending in 0, the next number is the smallest missing number. So if x is missing, find dm0 > n_x, then the next term would be (0)x = x, a contradiction. QED

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Example

a(2) = 1 since it is formed from a(1) = 0 as 01 = 1.
a(20) = 11 since it is the smallest number not yet in the sequence that starts with the ending digit 0 of a(19) = 90.

Mathematica

Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], If[# == 0, True, First@ IntegerDigits@ k == #] &@ Mod[#[[-1]], 10]], k++]; k]] &, {0}, 69] (* Michael De Vlieger, Oct 15 2018 *)

Prog

(PARI) nexta(v, x) = {my(d = x % 10, newa); for (i=0, oo, newa = eval(concat(Str(d), Str(i))); if (! vecsearch(v, newa), return (newa)); ); }
lista(nn) = {lasta = 0; print1(lasta, ", "); va = [lasta]; for (n=1, nn, newa = nexta(va, lasta); print1(newa, ", "); va = vecsort(concat(va, newa)); lasta = newa; ); } \\ Michel Marcus, Oct 15 2018

Crossrefs

Cf. A008592, A162501, A173237, A261370.

Sequence in context: A040099 A330365 A173237 * A086068 A262557 A030595

Adjacent sequences: A319151 A319152 A319153 * A319155 A319156 A319157

Keyword

nonn,base,look,hear

Author

Enrique Navarrete, Sep 25 2018

Status

approved