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A191106 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x are in a.

%I A191106 #57 Feb 16 2025 08:33:14
%S A191106 1,3,7,9,19,21,25,27,55,57,61,63,73,75,79,81,163,165,169,171,181,183,
%T A191106 187,189,217,219,223,225,235,237,241,243,487,489,493,495,505,507,511,
%U A191106 513,541,543,547,549,559,561,565,567,649,651,655,657,667,669,673,675,703,705,709,711,721,723,727,729,1459,1461,1465,1467,1477
%N A191106 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x are in a.
%C A191106 Related sequences for various choices of i and k as defined in A190803:
%C A191106   A003278:  (i,k) = (-2,-1)
%C A191106   A191106:  (i,k) = (-2, 0)
%C A191106   A191107:  (i,k) = (-2, 1)
%C A191106   A191108:  (i,k) = (-2, 2)
%C A191106   A153775:  (i,k) = (-1, 0)
%C A191106   A147991:  (i,k) = (-1, 1)
%C A191106   A191109:  (i,k) = (-1, 2)
%C A191106   A005836:  (i,k) = ( 0, 1)
%C A191106   A191110:  (i,k) = ( 0, 2)
%C A191106   A132140:  (i,k) = ( 1, 2)
%C A191106 For a=A191106, we have closure properties: the integers in (2+a)/3 comprise a; the integers in a/3 comprise a.
%C A191106 For k >= 1, m = a(i), 1 <= i <= 2^k seems to be m such that m/(3^k+1) is in the Cantor set (except that m = 0 and m = 3^k+1 do not appear). For k >= 2,  m = (a(i)-1)/2, 1 <= i <= 2^k seems to be m such that m/((3^k-1)/2) is in the Cantor set. - _Peter Munn_, Jul 06 2019
%C A191106 Every even number is the sum of two (possibly equal) terms. More specifically: terms a(1) through a(2^n) = 3^n sum to even numbers 2 times 1 through 3^n. Every even number is infinitely often the difference of two terms. Since the sequence is equal to 2*A005836(n) + 1, these properties follow immediately from similar properties of A005836 for every number. - _Aad Thoen_, Feb 17 2022
%C A191106 if A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+2*3^n), similar to A003278. - _Arie Bos_, Jul 26 2022
%H A191106 Ivan Neretin, <a href="/A191106/b191106.txt">Table of n, a(n) for n = 1..10000</a>
%H A191106 David Garth and Adam Gouge, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Garth/garth14.html">Affinely Self-Generating Sets and Morphisms</a>, Journal of Integer Sequences, 10 (2007) 1-13.
%H A191106 D. Jordan and R. Schayer <a href="https://math.psu.edu/mass/sites/default/files/reu2003/6.pdf">Rational points on the Cantor middle thirds set</a>, Penn State, REU 2003.
%H A191106 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CantorSet.html">Cantor Set</a>
%F A191106 a(n) = 2*A005836(n) + 1. - _Charles R Greathouse IV_, Sep 06 2011
%F A191106 a(n) = A005823(n) + 1. - _Vladimir Shevelev_, Dec 17 2012
%F A191106 a(n) = (A191108(n) + 1)/2. - _Peter Munn_, Jul 09 2019
%e A191106 1 -> 3 -> 7,9 -> 19,21,25,27 -> ...
%t A191106 h = 3; i = -2; j = 3; k = 0; f = 1; g = 9;
%t A191106 a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191106; regarding g, see note at A190803 *)
%t A191106 b = (a + 2)/3; c = a/3; r = Range[1, 900];
%t A191106 d = Intersection[b, r](* illustrates closure property *)
%t A191106 e = Intersection[c, r](* illustrates closure property *)
%t A191106 2 FromDigits[#, 3]&/@Tuples[{0, 1}, 7] + 1 (* _Vincenzo Librandi_, Jul 10 2019 *)
%Y A191106 Cf. A005823, A005836, A054591, A088917 (characteristic function), A173934, A190803, A191108.
%Y A191106 Partial sums of A061393.
%Y A191106 Similar formula as A003278, A_(n+1)=(A_n,A_n+2*3^n).
%K A191106 nonn,easy
%O A191106 1,2
%A A191106 _Clark Kimberling_, May 26 2011