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a(n) is the smallest number not already in the sequence, such that the concatenation of all a(n) displays the periodic digit string 1, 2, 3 (and repeat).
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%I #6 May 05 2019 02:14:04

%S 1,2,3,12,31,23,123,1231,231,2312,312,3123,12312,31231,23123,123123,

%T 1231231,231231,2312312,312312,3123123,12312312,31231231,23123123,

%U 123123123,1231231231,231231231,2312312312,312312312,3123123123,12312312312,31231231231

%N a(n) is the smallest number not already in the sequence, such that the concatenation of all a(n) displays the periodic digit string 1, 2, 3 (and repeat).

%e Starting from 1, 2, 3, the next number must be 12 because we need to continue with a 1. But 1 is already in the sequence so we need to attach a 2 -> 12. And so on.

%p cyc3 := proc(n) op(n,[2,3,1]) ; end:

%p A165301 := proc(n) option remember ; local k,prev,d,a ; if n = 1 then 1; else d := cyc3(procname(n-1) mod 10) ; a := d ; while true do prev := false; for k from 1 to n-1 do if procname(k) = a then prev := true; break; end if; end do; if not prev then return a; end if; d := cyc3(d) ; a := 10*a+d ; end do; end if ; end proc:

%p seq(A165301(n),n=1..60) ; # _R. J. Mathar_, Oct 16 2009

%Y Cf. A165300, A165302-A165307.

%K easy,base,nonn

%O 1,2

%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Sep 14 2009

%E Keyword:base added, sequence extended by _R. J. Mathar_, Oct 16 2009