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%I #38 Mar 12 2023 08:48:33
%S 1,111,11111,1111111,111111111,11111111111,1111111111111,
%T 111111111111111,11111111111111111,1111111111111111111,
%U 111111111111111111111,11111111111111111111111
%N Bisection of A002275.
%C 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
%C 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
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H H. C. Williams and R. K. Guy, <a href="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (101,-100).
%F Numbers composed entirely of 2*n+1 concatenated 1's for n >= 0.
%F O.g.f.: (1+10*x)/((-1+x)*(-1+100*x)). - _R. J. Mathar_, Apr 03 2008
%F From _Klaus Purath_, Sep 23 2020: (Start)
%F a(n) = Sum_{i = 0..2*n} 10^i.
%F a(n) = 101*a(n-1) - 100*a(n-2).
%F a(n) = 110*10^(2*n-2) + a(n-1).
%F a(n) = 100*a(n-1) + 11.
%F a(n) = (a(n-1)^2 - 1210*10^(2*n-4))/a(n-2). (End)
%p seq((10^(2*n+1) - 1)/9,n=0..15); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005
%t Table[(10^(2*n + 1) - 1)/9, {n, 0, 100}] (* _Robert Price_, Feb 21 2016 *)
%o (Python)
%o def A100706(n): return (10**((n<<1)+1)-1)//9 # _Chai Wah Wu_, Nov 04 2022
%o (PARI) a(n) = (10^(2*n + 1) - 1)/9; \\ _Michel Marcus_, Mar 12 2023
%Y Cf. A002275, A099814 (other bisection), A007583, A095372, A299960.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 19 2004
%E More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005
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