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%I #40 Dec 30 2022 03:54:54
%S 1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,3,1,2,
%T 1,3,1,2,1,1,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,1,1,2,1,3,
%U 1,2,1,1,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,3,1,2,1,3,1,2,1,1,1
%N Start with 1,2, concatenate 2^k previous terms and change last term as follows: 1->2, 2->3, 3->1.
%H Antti Karttunen, <a href="/A078734/b078734.txt">Table of n, a(n) for n = 1..20000</a> (first 1024 terms from Andrew Howroyd)
%F (Sum_{k=1..n} a(k))/n -> 1.57.... [This limit is the asymptotic mean of this sequence, 11/7. - _Amiram Eldar_, Oct 28 2022]
%F Multiplicative with a(2^e) = (e mod 3) + 1, a(p^e) = 1 for odd prime p. - _Andrew Howroyd_, Jul 31 2018
%F From _Antti Karttunen_, Dec 07 2021: (Start)
%F a(n) = 1 + A096271(n-1) = 1 + A010872(A007814(n)). [as above]
%F a(n) = A001511(A050985(n)).
%F (End)
%F Dirichlet g.f.: zeta(s)*(4^s+2^(s+1)+3)/(4^s+2^s+1). - _Amiram Eldar_, Dec 30 2022
%e Concatenating the first 2 terms 1,2 -> 1,2,1,2 and changing 2->3 gives the first 4 terms: 1,2,1,3.
%e Concatenating those first 4 terms ->1,2,1,3,1,2,1,3 and changing 3->1 gives the first 8 terms: 1,2,1,3,1,2,1,1.
%t a[n_] := Mod[IntegerExponent[n, 2], 3] + 1; Array[a, 100] (* _Amiram Eldar_, Oct 28 2022 *)
%o (PARI) seq(n)={my(v=[1,2]); while(#v < n, v=concat(v,v); v[#v] = v[#v] % 3 + 1); vector(n, i, v[i])} \\ _Andrew Howroyd_, Jul 31 2018
%o (PARI) a(n) = valuation(n, 2) % 3 + 1; \\ _Andrew Howroyd_, Jul 31 2018
%o (Magma) [Valuation(n, 2) mod 3 + 1: n in [1..100]]; // _Vincenzo Librandi_, Aug 01 2018
%Y Cf. A056832, A035263.
%K nonn,mult
%O 1,2
%A _Benoit Cloitre_, Dec 21 2002
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