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

1, 3, 7, 9, 19, 21, 25, 27, 55, 57, 61, 63, 73, 75, 79, 81, 163, 165, 169, 171, 181, 183, 187, 189, 217, 219, 223, 225, 235, 237, 241, 243, 487, 489, 493, 495, 505, 507, 511, 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

Offset: 1

Author

Clark Kimberling, May 26 2011

Keywords

• nonn
• easy

Comments

Related sequences for various choices of i and k as defined in A190803:

A003278: (i,k) = (-2,-1)

A191106: (i,k) = (-2, 0)

A191107: (i,k) = (-2, 1)

A191108: (i,k) = (-2, 2)

A153775: (i,k) = (-1, 0)

A147991: (i,k) = (-1, 1)

A191109: (i,k) = (-1, 2)

A005836: (i,k) = ( 0, 1)

A191110: (i,k) = ( 0, 2)

A132140: (i,k) = ( 1, 2)

For a=A191106, we have closure properties: the integers in (2+a)/3 comprise a; the integers in a/3 comprise a.

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

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

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

Examples

			1 -> 3 -> 7,9 -> 19,21,25,27 -> ...
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.
• D. Jordan and R. Schayer Rational points on the Cantor middle thirds set, Penn State, REU 2003.
• Eric Weisstein's World of Mathematics, Cantor Set

Crossrefs

Cf. A005823, A005836, A054591, A088917 (characteristic function), A173934, A190803, A191108.

Partial sums of A061393.

Similar formula as A003278, A_(n+1)=(A_n,A_n+2*3^n).

Programs

Mathematica

h = 3; i = -2; j = 3; k = 0; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191106; regarding g, see note at A190803 *)
b = (a + 2)/3; c = a/3; r = Range[1, 900];
d = Intersection[b, r](* illustrates closure property *)
e = Intersection[c, r](* illustrates closure property *)
2 FromDigits[#, 3]&/@Tuples[{0, 1}, 7] + 1 (* Vincenzo Librandi, Jul 10 2019 *)

Formula

a(n) = 2*A005836(n) + 1. - Charles R Greathouse IV, Sep 06 2011

a(n) = A005823(n) + 1. - Vladimir Shevelev, Dec 17 2012

a(n) = (A191108(n) + 1)/2. - Peter Munn, Jul 09 2019