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

Author

Clark Kimberling, May 26 2011

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

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  *)

PARI

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

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)