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%I #32 Jan 04 2023 02:04:48
%S 1,2,1,3,1,2,1,0,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,0,1,2,1,3,1,2,1,2,1,2,
%T 1,3,1,2,1,0,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,0,1,2,1,3,1,2,1,3,1,2,1,3,
%U 1,2,1,0,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,0,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,0
%N Fixed point of the morphism 0->11, 1->12, 2->13, 3->10, starting from a(1) = 1.
%H Andrew Howroyd, <a href="/A092412/b092412.txt">Table of n, a(n) for n = 1..1024</a>
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>.
%F a(n) = A001511(n) mod 4.
%F a(2n+1) = 1; a(2n) = a(n) + 1 mod 4.
%F a(n) == A035263(n) (mod 2); a(n) == A033485(n) (mod 2).
%F Multiplicative with a(2^e) = (1 + e) mod 4, a(p^e) = 1 for odd prime p. - _Andrew Howroyd_, Aug 06 2018
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 22/15. - _Amiram Eldar_, Nov 29 2022
%F Dirichlet g.f.: zeta(s)*(3*2^s+2^(2*s+1)+2^(3*s))/(1+2^s+4^s+8^s). - _Amiram Eldar_, Jan 04 2023
%t Nest[ Function[ l, {Flatten[(l /. {0 -> {1, 1}, 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1, 0}})] }], {0}, 7] (* _Robert G. Wilson v_, Mar 04 2005 *)
%t SubstitutionSystem[{0 -> {1, 1}, 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1, 0}}, {1}, 7] // Last (* _Jean-François Alcover_, Sep 20 2019 *)
%o (PARI) a(n)=(1 + valuation(n, 2)) %4; \\ _Andrew Howroyd_, Aug 06 2018
%o (Python)
%o def A092412(n): return (n&-n).bit_length()&3 # _Chai Wah Wu_, Jul 13 2022
%Y Cf. A001511, A033485, A035263, A346070 (same with values 3,0,1,2).
%K easy,mult,nonn
%O 1,2
%A _Philippe Deléham_, Mar 22 2004
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