A267045 -id:A267045 - cp's OEIS frontend

A267044 Binary representation of the middle column of the "Rule 91" elementary cellular automaton starting with a single ON (black) cell.

1, 10, 100, 1000, 10001, 100010, 1000101, 10001010, 100010101, 1000101010, 10001010101, 100010101010, 1000101010101, 10001010101010, 100010101010101, 1000101010101010, 10001010101010101, 100010101010101010, 1000101010101010101, 10001010101010101010

Offset: 0

Robert Price, Jan 09 2016

nonn

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A267015, A267045.

rule=91; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) mc=Table[catri[[k]][[k]],{k,1,rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc,k]],{k,1,rows}] (* Binary Representation of Middle Column *)

Conjectures from Colin Barker, Jan 10 2016: (Start)
a(n) = (-550+450*(-1)^n+9901*10^n)/9900 for n>1.
a(n) = 10*a(n-1)+a(n-2)-10*a(n-3) for n>4. [Typo corrected by Karl V. Keller, Jr., Mar 16 2022]
G.f.: (1-x^2+x^4) / ((1-x)*(1+x)*(1-10*x)).
(End)

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - Michael De Vlieger, Jun 13 2022

A294523 Lexicographically earliest sequence of positive terms, such that, for any n > 0, the binary expansion of n, say of size k+1, is (1, a(n) mod 2, a^2(n) mod 2, ..., a^k(n) mod 2) (where a^i denotes the i-th iterate of the sequence).

Original entry on oeis.org

1, 2, 1, 2, 6, 5, 1, 2, 10, 6, 14, 9, 5, 13, 1, 2, 18, 10, 22, 12, 6, 14, 30, 17, 9, 5, 11, 25, 13, 29, 1, 2, 34, 18, 38, 20, 10, 22, 46, 24, 12, 6, 54, 28, 14, 30, 62, 33, 17, 9, 19, 41, 5, 11, 23, 49, 25, 13, 27, 57, 29, 61, 1, 2, 66, 34, 70, 36, 18, 38, 78

Offset: 1

Rémy Sigrist, Nov 01 2017

More informally, the parity of the iterate of the sequence at n gives the binary expansion of n (beyond the leading 1).
Apparently, iterating the sequence always leads to one of these three loops:
- the fixed point (1) iff we start from 2^k-1 for some k > 0,
- the fixed point (2) iff we start from 2^k for some k > 0,
- or (5, 6) for any other starting value.
a(n) is even iff n belongs to A004754.
a(n) is odd iff n belongs to A004760.
If a(n) > n then a(n) = A080541(n).
If n < 2^k then a(n) < 2^k.
Apparently, if a(n) > 2, then A054429(a(n)) = a(A054429(n)); this accounts for the symmetry of the part connected to the loop (5,6) in the oriented graph of this sequence.

			For n=11:
- the binary representation of 11 is (1,0,1,1),
- a(11) = 14 has parity 0,
- a(14) = 13 has parity 1,
- a(13) = 5 has parity 1,
- we find the binary digits of 11 beyond the initial 1, in order: 0, 1, 1.
See also representations of first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, Representation of the first 32 terms as an oriented graph (n -> a(n))
Rémy Sigrist, Representation of the first 64 terms as an oriented graph (n -> a(n))
Rémy Sigrist, PARI program for A294523

Cf. A000079, A000225, A000975, A004754, A004760, A052997, A054429, A080541, A081253, A081254, A266248, A266613, A266721, A267045.

PARI
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See Links section.
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a(n) = 1 iff n = A000225(k) for some k > 0.
a(n) = 2 iff n = A000079(k) for some k > 0.
a(n) = 5 iff n = A081254(k) for some k > 2.
a(n) = 6 iff n = A000975(k) for some k > 2.
a(n) = 10 iff n = A081253(k) for some k > 2.
a(n) = 12 iff n = A266613(k) for some k > 3.
a(n) = 13 iff n = A052997(k) for some k > 2.
a(n) = 14 iff n = A266721(k) for some k > 2.
a(n) = 18 iff n = A267045(k) for some k > 3.
a(n) = 54 iff n = A266248(k) for some k > 4.
These formulas come from the fact that each sequence on the right side, say f, eventually satisfies: f(n) = floor(f(n+1)/2), and f(n) and f(n+2) have the same parity.

cp's OEIS Frontend

A267044 Binary representation of the middle column of the "Rule 91" elementary cellular automaton starting with a single ON (black) cell.

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