A100706 Bisection of A002275.

1, 111, 11111, 1111111, 111111111, 11111111111, 1111111111111, 111111111111111, 11111111111111111, 1111111111111111111, 111111111111111111111, 11111111111111111111111

Offset: 0

Author

N. J. A. Sloane, Nov 19 2004

Keywords

• nonn
• easy

Comments

Also the binary representation of the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell for Rules 151, 159, 183, 191, 215, 222, 223, 247, 254 and 255. - Robert Price, Feb 21 2016

The aerated sequence 1, 0, 111, 0, 11111, 0, 1111111, ... is a linear divisibility sequence of order 4. It is the case P1 = 0, P2 = -9^2, Q = -10 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. Cf. A007583, A095372 and A299960. - Peter Bala, Aug 28 2019

References

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

Links

• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
• S. Wolfram, A New Kind of Science
• Index to Elementary Cellular Automata
• Index entries for sequences related to cellular automata
• Index entries for linear recurrences with constant coefficients, signature (101,-100).

Crossrefs

Cf. A002275, A099814 (other bisection), A007583, A095372, A299960.

Programs

Maple

seq((10^(2*n+1) - 1)/9,n=0..15); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

Mathematica

Table[(10^(2*n + 1) - 1)/9, {n, 0, 100}] (* Robert Price, Feb 21 2016 *)

PARI

a(n) = (10^(2*n + 1) - 1)/9; \\ Michel Marcus, Mar 12 2023

Python

def A100706(n): return (10**((n<<1)+1)-1)//9 # Chai Wah Wu, Nov 04 2022

Formula

Numbers composed entirely of 2*n+1 concatenated 1's for n >= 0.

O.g.f.: (1+10*x)/((-1+x)*(-1+100*x)). - R. J. Mathar, Apr 03 2008

From Klaus Purath, Sep 23 2020: (Start)

a(n) = Sum_{i = 0..2*n} 10^i.

a(n) = 101*a(n-1) - 100*a(n-2).

a(n) = 110*10^(2*n-2) + a(n-1).

a(n) = 100*a(n-1) + 11.

a(n) = (a(n-1)^2 - 1210*10^(2*n-4))/a(n-2). (End)

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005