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

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, 43, 44, 51, 76, 83, 84, 85, 85, 85, 86, 89, 102, 153, 166, 169, 170, 170, 170, 171, 172, 179, 204, 307, 332, 339, 340, 341, 341, 341, 342, 345, 358, 409, 614, 665

Offset: 0

Author

Raphael J. F. Berger, Jan 27 2023

Keywords

• nonn
• base

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 right half here is the part which began as 0's.

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.

Examples

			Following the state progression from A359303 (state(n)) is converted to the sequence (a(n)) by:
               state(0) =   ..1111|0000..
                                  |0000..
                       a(0) = 0 = \---> bits 000..
               state(1) =   ..1110|1000..
                                  |1000..
                       a(1) = 1 = \---> bits 100..
               state(2) = ..111101|10000..
                                  |10000..
                       a(2) = 1 = \---> bits 100..
               state(3) = ..111101|01000..
                                  |01000..
                       a(3) = 2 = \---> bits 01000..
               state(4) = ..111011|01000..
                       a(4) = 2 = \---> bits 01000..
               state(5) = ..111010|11000..
                       a(5) = 3 = \---> bits 11000..
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..3000
• Kevin Ryde, PARI/GP Code

Crossrefs

Cf. Base sequence A359303. Encoding of complementary left-half in A360142.

Programs

Mathematica

ClearAll[{s, prop, checkprop, doprop, p, a, j,runpos}];
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]]]
(* show only positive states: *)
runpos[n_]:=( s=Join[Array[#/#&, n+5], Array[0#&, n+5]] ; Table[Drop[Nest[doprop[#]&, s, k],n+5], {k, 0, n}])
(* conversion from the automaton states to integers *)
(* a[10] returns {0,1,1,2,2,3,4,5,5,6,9} *)
a[j_]:=Table[FromDigits[Reverse[runpos[j+1][[k, All]]],2], {k, 1, j+1}]

PARI

\\ See links.