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A360141 Bitwise encoding of the right half, initially empty, state of the 1D cellular automaton from A359303 after n steps.

%I A360141 #36 Dec 19 2024 11:46:19
%S A360141 0,1,1,2,2,3,4,5,5,6,9,10,10,11,12,19,20,21,21,21,22,25,38,41,42,42,
%T A360141 43,44,51,76,83,84,85,85,85,86,89,102,153,166,169,170,170,170,171,172,
%U A360141 179,204,307,332,339,340,341,341,341,342,345,358,409,614,665
%N A360141 Bitwise encoding of the right half, initially empty, state of the 1D cellular automaton from A359303 after n steps.
%C A360141 See A359303 for how the automaton steps.
%C A360141 The automaton state is a bi-infinite string of 1's and 0's of the form ...1111 middle 0000... and the right half here is the part which began as 0's.
%C A360141 The right half state is encoded in an integer by interpreting the bits from left to right as binary from least to most significant bit.
%H A360141 Kevin Ryde, <a href="/A360141/b360141.txt">Table of n, a(n) for n = 0..3000</a>
%H A360141 Kevin Ryde, <a href="/A360141/a360141.gp.txt">PARI/GP Code</a>
%e A360141 Following the state progression from A359303 (state(n)) is converted to the sequence (a(n)) by:
%e A360141                state(0) =   ..1111|0000..
%e A360141                                   |0000..
%e A360141                        a(0) = 0 = \---> bits 000..
%e A360141                state(1) =   ..1110|1000..
%e A360141                                   |1000..
%e A360141                        a(1) = 1 = \---> bits 100..
%e A360141                state(2) = ..111101|10000..
%e A360141                                   |10000..
%e A360141                        a(2) = 1 = \---> bits 100..
%e A360141                state(3) = ..111101|01000..
%e A360141                                   |01000..
%e A360141                        a(3) = 2 = \---> bits 01000..
%e A360141                state(4) = ..111011|01000..
%e A360141                        a(4) = 2 = \---> bits 01000..
%e A360141                state(5) = ..111010|11000..
%e A360141                        a(5) = 3 = \---> bits 11000..
%t A360141 ClearAll[{s, prop, checkprop, doprop, p, a, j,runpos}];
%t A360141 prop[s_]:=(p=Array[0#&, Length[s]];
%t A360141 Do[If[i==1 ||i==Length[s], p[[i]]=0,
%t A360141 {p[[i-1]], p[[i]], p[[i+1]]}+=
%t A360141 Piecewise[{{{1, -1, 0}, {s[[i-1]], s[[i]], s[[i+1]]}=={0, 1, 1}},
%t A360141 {{0, -1, 1}, {s[[i-1]], s[[i]], s[[i+1]]}=={1, 1, 0}}}, {0, 0, 0}]], {i, 1, Length[s]-1} ];
%t A360141 Return[p])
%t A360141 checkprop[s_]:=(p=s;
%t A360141 Do[If[p[[i]]==2, {p[[i-1]], p[[i]], p[[i+1]]}={0, 0, 0}], {i, 2, Length[s]-1}];
%t A360141 Return[p])
%t A360141 doprop[s_]:= Return[s +checkprop[prop[s]]]
%t A360141 (* show only positive states: *)
%t A360141 runpos[n_]:=( s=Join[Array[#/#&, n+5], Array[0#&, n+5]] ; Table[Drop[Nest[doprop[#]&, s, k],n+5], {k, 0, n}])
%t A360141 (* conversion from the automaton states to integers *)
%t A360141 (* a[10] returns {0,1,1,2,2,3,4,5,5,6,9} *)
%t A360141 a[j_]:=Table[FromDigits[Reverse[runpos[j+1][[k, All]]],2], {k, 1, j+1}]
%o A360141 (PARI) \\ See links.
%Y A360141 Cf. Base sequence A359303. Encoding of complementary left-half in A360142.
%K A360141 nonn,base
%O A360141 0,4
%A A360141 _Raphael J. F. Berger_, Jan 27 2023