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A191108 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x+2 are in a.

%I A191108 #26 Feb 16 2025 08:33:14
%S A191108 1,5,13,17,37,41,49,53,109,113,121,125,145,149,157,161,325,329,337,
%T A191108 341,361,365,373,377,433,437,445,449,469,473,481,485,973,977,985,989,
%U A191108 1009,1013,1021,1025,1081,1085,1093,1097,1117,1121,1129,1133,1297,1301,1309,1313,1333,1337,1345,1349,1405,1409,1417,1421,1441,1445
%N A191108 Increasing sequence generated by these rules:  a(1)=1, and if x is in a then 3x-2 and 3x+2 are in a.
%C A191108 See discussions at A190803, A191106.  The sequence a=A191108 has closure properties:  the positive integers in (2+A191108)/3 comprise A191108, as do those in (-2+A191108)/3.
%C A191108 From _Peter Munn_, May 13 2019: (Start)
%C A191108 The closure of {1} in the positive integers under reflection about 3^k, k >= 1.
%C A191108 Asymptotic density is 0.
%C A191108 Consider a Sierpinski arrowhead curve formed of edges numbered consecutively from 0 at its axis of symmetry. The m-th edge is contained in the boundary of the plane sector occupied by the arrowhead if and only if m or -m is in this sequence.
%C A191108 For k >= 0, a(2^k) = 2*3^k - 1 and {a(i)/(2*3^k) | 1 <= i <= 2^k} is the set of center points of surviving intervals at the k-th step of generating the Cantor set, and therefore the set of center points of deleted middle-third intervals at the (k+1)-th step.
%C A191108 Define t: Z -> P(R) so that t(n) is the translated Cantor ternary set spanning [(n-1)/2, (n+1)/2], and let T be the union of t(a(n)) for all n. T = T * 3 = T / 3 is the closure of the Cantor ternary set under multiplication by 3.
%C A191108 (End)
%H A191108 Ivan Neretin, <a href="/A191108/b191108.txt">Table of n, a(n) for n = 1..10000</a>
%H A191108 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CantorSet.html">Cantor Set</a>
%H A191108 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sierpi%C5%84ski_arrowhead_curve">Sierpiński arrowhead curve</a>
%F A191108 From _Peter Munn_, May 25 2019: (Start)
%F A191108 a(n) = (A055247(2n-1) + A055247(2n)) / 3.
%F A191108 a(n) = A306556(2n)*2 - 1 = A306556(2n-1) + A306556(2n).
%F A191108 a(n) = 2*A005823(n) + 1 = 4*A005836(n) + 1 = 2*A191106(n) - 1.
%F A191108 a(2^k+i) = 2*A147991(2^k+i-1) + 3^(k+1) for k >= 0, 1 <= i <= 2^k.
%F A191108 (End)
%t A191108 h = 3; i = -2; j = 3; k = 2; f = 1; g = 7;
%t A191108 a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191108 *)
%t A191108 b = (a + 2)/3; c = (a - 2)/3; r = Range[1, 900];
%t A191108 d = Intersection[b, r] (* A191108 closure property  *)
%t A191108 e = Intersection[c, r] (* A191108 closure property  *)
%o A191108 (PARI) a(n) = fromdigits(binary(n-1),3)<<2 + 1; \\ _Kevin Ryde_, Aug 05 2022
%Y A191108 Cf. A005823, A005836, A055247, A147991, A190803, A191106, A306556.
%K A191108 nonn,easy
%O A191108 1,2
%A A191108 _Clark Kimberling_, May 26 2011