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.

1, 5, 13, 17, 37, 41, 49, 53, 109, 113, 121, 125, 145, 149, 157, 161, 325, 329, 337, 341, 361, 365, 373, 377, 433, 437, 445, 449, 469, 473, 481, 485, 973, 977, 985, 989, 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

Offset

1,2

Comments

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.

From Peter Munn, May 13 2019: (Start)

The closure of {1} in the positive integers under reflection about 3^k, k >= 1.

Asymptotic density is 0.

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.

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.

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.

(End)

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Cantor Set

Wikipedia, Sierpiński arrowhead curve

Formula

From Peter Munn, May 25 2019: (Start)

a(n) = (A055247(2n-1) + A055247(2n)) / 3.

a(n) = A306556(2n)*2 - 1 = A306556(2n-1) + A306556(2n).

a(n) = 2*A005823(n) + 1 = 4*A005836(n) + 1 = 2*A191106(n) - 1.

a(2^k+i) = 2*A147991(2^k+i-1) + 3^(k+1) for k >= 0, 1 <= i <= 2^k.

(End)

Mathematica

h = 3; i = -2; j = 3; k = 2; f = 1; g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191108 *)
b = (a + 2)/3; c = (a - 2)/3; r = Range[1, 900];
d = Intersection[b, r] (* A191108 closure property  *)
e = Intersection[c, r] (* A191108 closure property  *)

Prog

(PARI) a(n) = fromdigits(binary(n-1), 3)<<2 + 1; \\ Kevin Ryde, Aug 05 2022

Crossrefs

Cf. A005823, A005836, A055247, A147991, A190803, A191106, A306556.

Sequence in context: A184851 A211425 A245906 * A216575 A306626 A053028

Adjacent sequences: A191105 A191106 A191107 * A191109 A191110 A191111

Keyword

nonn,easy

Author

Clark Kimberling, May 26 2011

Status

approved