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Sequence S such that 1 is in S and if x is in S, then 3x-1 and 3x are in S.
8

%I #18 May 08 2022 15:59:05

%S 1,2,3,5,6,8,9,14,15,17,18,23,24,26,27,41,42,44,45,50,51,53,54,68,69,

%T 71,72,77,78,80,81,122,123,125,126,131,132,134,135,149,150,152,153,

%U 158,159,161,162,203,204,206,207,212,213,215,216,230,231,233,234,239,240,242

%N Sequence S such that 1 is in S and if x is in S, then 3x-1 and 3x are in S.

%C Subsequences include A007051, A000244, A153773, A153774.

%C First generation: 1

%C 2nd generation: 2, 3

%C 3rd generation: 5, 6, 8, 9

%C 4th generation: 14, 15, 17, 18, 23, 24, 26, 27

%C Does every generation contain a prime?

%C From _Peter Munn_, Feb 10 2022: (Start)

%C Consider a Sierpinski arrowhead curve formed of edges indexed consecutively from 0 at its axis of symmetry and aligned with an infinite Sierpinski gasket so that each edge is contained in the boundary of either the plane sector occupied by the gasket or a triangular region of the gasket's complement. The numbers {4*a(n)-1 : n >= 1} (that is, 3, 7, 11, 19, 23, 31, 35, 55, 59, ...) index the edges that are contained in the boundaries of certain triangular regions: each is the first region encountered of each successively larger size that does not lie across the axis of symmetry.

%C Let S be the set of terms. Define c: N -> P(R) so that c(m) is the scaled and translated Cantor ternary set spanning [m-0.5, m], and let C be the union of c(m) for all m in S. C is the closure under multiplication by 3 of the scaled and translated Cantor ternary set spanning [0.5, 1.0].

%C (End)

%C Positive numbers whose balanced ternary expansions contain exactly one digit 1. - _Rémy Sigrist_, May 08 2022

%H G. C. Greubel, <a href="/A153775/b153775.txt">Table of n, a(n) for n = 1..1000</a>

%H Peter Munn, <a href="/A309054/a309054_1.png">Edges of Sierpiński arrowhead aligned with Sierpiński gasket</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CantorSet.html">Cantor Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Closure.html">Closure</a>

%F From _Peter Munn_, Feb 04 2022: (Start)

%F For k >= 0, 2^k <= n <= 2^(k+1)-1, a(n) = A005836(n+1) - (3^k-1)/2.

%F For n >= 1, A307744(4*a(2n)-1) = A307744(4*a(2n+1)-1) = A307744(4*a(n)-1) + 1.

%F (End)

%t nxt[n_] := Flatten[3 # + {-1, 0} & /@ n]; Union[Flatten[NestList[nxt,{1},5]]] (* _G. C. Greubel_, Aug 28 2016 *)

%Y Cf. A000244, A007051, A005836, A147992, A153773, A153774, A306556, A307744.

%Y See also the related sequences listed in A191106.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 02 2009