A360142 Bitwise encoding of the left half, initially fully occupied, state of the 1D cellular automaton from A359303 after n steps.

0, 1, 2, 2, 4, 5, 8, 9, 10, 17, 18, 18, 20, 35, 36, 37, 40, 69, 73, 74, 81, 138, 145, 146, 146, 148, 163, 276, 291, 292, 293, 296, 325, 553, 582, 585, 586, 593, 650, 1105, 1162, 1169, 1170, 1172, 1187, 1300, 2211, 2324, 2339, 2340, 2341, 2344, 2373, 2601

Offset

0,3

Comments

See A359303 for how the automaton steps.

The automaton state is a bi-infinite string of 1's and 0's of the form ...1111 middle 0000... and the left half here is the part which began as 1's.

The left half state is encoded in an integer by inverting the bits (0<->1) and interpreting the them from right to left as binary from least to most significant bit.

Links

Kevin Ryde, Table of n, a(n) for n = 0..3000

Kevin Ryde, PARI/GP Code

Example

Following the state progression from A359303 (state(n)) is converted to the sequence (a(n)) by:
               state(0) =   ..1111|0000..
                            ..1111|
                            ..0000|
   a(0) = 0 = bits               0
               state(1) =   ..1110|1000..
                            ..1110|
                            ..0001|
   a(1) = 1 = bits               1
               state(2) = ..111101|10000..
                          ..111101|
                          ..000010|
   a(2) = 2 = bits              10
               state(3) = ..111101|10000..
                          ..111101|
                          ..000010|
   a(3) = 2 = bits              10
               state(4) = ..111011|01000..
                          ..111011|
                          ..000100|
   a(4) = 4 = bits             100
               state(5) = ..111010|11000..
                          ..111010|
                          ..000101|
   a(5) = 5 = bits             101

Mathematica

ClearAll[{s, prop, checkprop, doprop, run, p, a, j, runneg}];
prop[s_]:=(p=Array[0#&, Length[s]];
Do[If[i==1 ||i==Length[s], p[[i]]=0,
{p[[i-1]], p[[i]], p[[i+1]]}+=
Piecewise[{{{1, -1, 0}, {s[[i-1]], s[[i]], s[[i+1]]}=={0, 1, 1}},
{{0, -1, 1}, {s[[i-1]], s[[i]], s[[i+1]]}=={1, 1, 0}}}, {0, 0, 0}]], {i, 1, Length[s]-1} ];
Return[p])
checkprop[s_]:=(p=s;
Do[If[p[[i]]==2, {p[[i-1]], p[[i]], p[[i+1]]}={0, 0, 0}], {i, 2, Length[s]-1}];
Return[p])
doprop[s_]:= Return[s +checkprop[prop[s]]]
runneg[n_]:=( s=Join[Array[#/#&, n+5], Array[0#&, n+5]] ; Table[Drop[Nest[doprop[#]&, s, k], -(n+5)], {k, 0, n}])
a[j_]:=FromDigits[(runneg[j+1]/.{0->1, 1->0})[[j+1, All]], 2]
(* Table[a[n], {n, 0, 10, 1}]          *)
(* returns the first 11 elements   *)
(* {0, 1, 2, 2, 4, 5, 8, 9, 10, 17, 18}      *)

Prog

(PARI) See links.

Crossrefs

Cf. A359303, A360141.

Sequence in context: A308902 A166515 A339560 * A308952 A302401 A326443

Adjacent sequences: A360139 A360140 A360141 * A360143 A360144 A360145

Keyword

nonn,base

Author

Raphael J. F. Berger, Jan 27 2023

Status

approved