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A365197 a(n) and a(n+1) have k distinct digits in common. The successive ks are the successive digits of the sequence itself.

%I A365197 #19 Sep 03 2023 10:47:06
%S A365197 1,10,12,3,13,103,130,124,142,2,104,140,123,132,4,14,1024,1042,1356,
%T A365197 1365,15,51,5,1023,1032,1456,1465,7,17,107,170,125,152,1026,1062,1345,
%U A365197 1354,16,20,1025,1052,18,2034,2043,23,102,10234,102345,102354,167,102367,102376
%N A365197 a(n) and a(n+1) have k distinct digits in common. The successive ks are the successive digits of the sequence itself.
%C A365197 All terms are distinct and composed by distinct digits. This is the lexicographically earliest sequence with this property. The sequence is finite but its last term is not known by the author. Note that the sequence would stop much earlier (with a(6) = 31) without an extra rule: a(n) has equal or more digits than the n-th digit of the sequence. Extra rule by _Giorgos Kalogeropoulos_.
%H A365197 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2023/08/shared-digits.html">Shared digits</a>, personal blog.
%e A365197 a(1) = 1 and a(2) = 10 share exactly 1 digit;
%e A365197 a(2) = 10 and a(3) = 12 share exactly 1 digit;
%e A365197 a(3) = 12 and a(4) =  3 share 0 digit;
%e A365197 a(4) = 3 and a(5) = 13 share exactly 1 digit;
%e A365197 a(5) = 13 and a(6) = 103 share exactly 2 digits, etc.
%e A365197 We see that the successive numbers of shared digits are the successive digits of the sequence itself (the successive shared digits 1, 1, 0, 1, 2 are the successive digits of the integers that start the sequence: 1, 10, 12).
%t A365197 a[1]=1;a[n_]:=a[n]=(c=1;While[!DuplicateFreeQ[in=(i=IntegerDigits)@c]|| MemberQ[ar=Array[a,n-1],c]||Length@Intersection[i@a[n-1],in]!=Flatten[i/@ar][[n-1]]||IntegerLength@c<Flatten[i/@ar][[n]],c++];c);Array[a,50] (* _Giorgos Kalogeropoulos_, Aug 25 2023 *)
%Y A365197 Cf. A184992.
%K A365197 base,nonn,fini
%O A365197 1,2
%A A365197 _Eric Angelini_, Aug 25 2023