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For any n > 0, let w be the least positive number such that the values (a(n+1-w), ..., a(n-1), e) do not appear continuously in (a(1), ..., a(n-1)) for some e in 0..w-1; a(n) is the least such e.
3

%I #6 May 19 2020 19:15:28

%S 0,0,1,0,0,0,2,0,0,0,0,1,1,0,1,2,1,0,2,1,1,1,2,0,1,0,1,0,2,2,0,2,0,1,

%T 1,1,0,0,1,2,2,1,2,0,0,1,3,0,0,0,3,1,0,0,2,1,0,0,3,0,1,0,3,2,0,0,2,2,

%U 2,0,0,3,2,1,0,1,1,2,1,1,0,2,0,2,1,2,1

%N For any n > 0, let w be the least positive number such that the values (a(n+1-w), ..., a(n-1), e) do not appear continuously in (a(1), ..., a(n-1)) for some e in 0..w-1; a(n) is the least such e.

%C This sequence is an unbounded variant of A334941.

%C Will every finite sequence of nonnegative integers appear?

%H Rémy Sigrist, <a href="/A334944/a334944.png">Colored scatterplot of the ordinal transform of the first 1000000 terms</a>

%H Rémy Sigrist, <a href="/A334944/a334944.pl.txt">Perl program for A334944</a>

%e For n = 1:

%e - for w = 1: (0) did not appear,

%e - so a(1) = 0.

%e For n = 2:

%e - for w = 1: (0) appeared,

%e - for w = 2: (0, 0) did not appear,

%e - so a(2) = 0.

%e For n = 3:

%e - for w = 1: (0) appeared,

%e - for w = 2: (0, 0) appeared but (0, 1) did not,

%e - so a(3) = 1.

%o (Perl) See Links section.

%Y See A334941 and A334956 for similar sequences.

%K nonn

%O 1,7

%A _Rémy Sigrist_, May 17 2020