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A374646 Paradiddle version of Thue-Morse sequence.

%I A374646 #45 Jan 11 2025 04:46:50
%S A374646 1,0,1,1,0,1,0,0,1,0,1,1,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0,1,0,0,1,0,
%T A374646 1,1,0,1,0,0,1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,1,0,1,1,1,0,1,1,0,1,0,0,
%U A374646 1,0,1,1,0,1,0,0,0,1,0,0,1,0,1,1
%N A374646 Paradiddle version of Thue-Morse sequence.
%C A374646 A paradiddle is a basic drum pattern, either "left left right left" or "right right left right". We can take left, right to be either 0, 1 or 1, 0.
%C A374646 Limiting word of the morphism with maps 0 |--> 0100, 1 |--> 1011 and axiom 1011. - _Joerg Arndt_, Jul 15 2024
%H A374646 Robert P. P. McKone, <a href="/A374646/b374646.txt">Table of n, a(n) for n = 0..16383</a>
%H A374646 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F A374646 a(n) = A160381(n)+1 mod 2. - _Kevin Ryde_, Dec 28 2024
%e A374646 k = 0: Sequence starts at its simplest form;
%e A374646 1.
%e A374646 -----------------------------------------------
%e A374646 k = 1: The 1 of the initial sequence expands following the morphism rules, where 1 -> {1, 0, 1, 1} and 0 -> {0, 1, 0, 0}, resulting in;
%e A374646 1, 0, 1, 1.
%e A374646 -----------------------------------------------
%e A374646 k = 2: Each element of the initial sequence expands following the morphism rules, where 1 -> {1, 0, 1, 1} and 0 -> {0, 1, 0, 0};
%e A374646 1, 0, 1, 1,
%e A374646 0, 1, 0, 0,
%e A374646 1, 0, 1, 1,
%e A374646 1, 0, 1, 1.
%e A374646 -----------------------------------------------
%e A374646 k = 3: The expansion is applied recursively, giving:
%e A374646 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1,
%e A374646 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0,
%e A374646 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1,
%e A374646 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1.
%t A374646 SubstitutionSystem[{1 -> {1, 0, 1, 1}, 0 -> {0, 1, 0, 0}}, {1}, {4}] // Flatten
%o A374646 (PARI) first(n,v=[1])=if(n>4*#v, v=first((n+3)\4)); my(u=List()); for(i=1,#v-1, listput(u,v[i]); listput(u,1-v[i]); listput(u,v[i]); listput(u,v[i])); my(t=vector(n-#u,i,if(i==2,1-v[#v],v[#v]))); for(j=1,#t, listput(u,t[j])); Vec(u) \\ _Charles R Greathouse IV_, Jul 31 2024
%Y A374646 Cf. A160381, A130198 (single paradiddle), A010059, A010060, A374724.
%K A374646 nonn,easy
%O A374646 0,1
%A A374646 _Robert P. P. McKone_, Jul 15 2024