A266248 -id:A266248 - cp's OEIS frontend

A266247 Binary representation of the middle column of the "Rule 9" elementary cellular automaton starting with a single ON (black) cell.

1, 10, 101, 1010, 10101, 101011, 1010110, 10101101, 101011010, 1010110101, 10101101010, 101011010101, 1010110101010, 10101101010101, 101011010101010, 1010110101010101, 10101101010101010, 101011010101010101, 1010110101010101010, 10101101010101010101

Offset: 0

Robert Price, Dec 25 2015

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

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (10,1,-10).

Cf. A266243, A266248.

rule=9; 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 *)

print([(100000*10**n//9 + 100001*10**n)//110000 for n in range(50)]) # Karl V. Keller, Jr., Dec 15 2021

From Colin Barker, Dec 28 2015 and Apr 14 2019: (Start)
a(n) = (-45000*(-1)^n + 1000009*10^n - 55000)/990000 for n > 3.
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n > 6.
G.f.: (1 + x^5 - x^6) / ((1-x)*(1+x)*(1-10*x)).
(End)
a(n) = floor((100000*10^n/9 + 100001*10^n)/110000). - Karl V. Keller, Jr., Dec 15 2021

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
```
See Links section.
```

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

A266247 Binary representation of the middle column of the "Rule 9" elementary cellular automaton starting with a single ON (black) cell.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Python

Formula