A191107 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x+1 are in a.

1, 4, 10, 13, 28, 31, 37, 40, 82, 85, 91, 94, 109, 112, 118, 121, 244, 247, 253, 256, 271, 274, 280, 283, 325, 328, 334, 337, 352, 355, 361, 364, 730, 733, 739, 742, 757, 760, 766, 769, 811, 814, 820, 823, 838, 841, 847, 850, 973, 976, 982, 985, 1000, 1003, 1009, 1012, 1054, 1057, 1063, 1066, 1081, 1084, 1090, 1093, 2188

Offset: 1

Author

Clark Kimberling, May 26 2011

Keywords

• nonn
• base
• easy

Comments

For general discussions, see A190803 and A191106.

Numbers whose base-3 representation ends in 1 and contains no 2; primitive members of A005836. - Peter Munn, Aug 14 2023

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.

Crossrefs

Cf. A003278, A005836, A055246, A190640, A190803, A191106.

Programs

Maple

N:= 100000: # to get all terms <= N
with(queue):
Q:= new(1):
A:= {}:
while not empty(Q) do
  s:= dequeue(Q);
  A:= A union {s};
  for t in {3*s-2,3*s+1} minus A do
    if t <= N then enqueue(Q,t) fi
  od
od:
sort(convert(A,list)); # Robert Israel, Nov 29 2015

Mathematica

h = 3; i = -2; j = 3; k = 1; f = 1;  g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191107 *)
b = (a + 2)/3; c = (a - 1)/3; r = Range[1, 900];
d = Intersection[b, r] (* A003278 *)
e = Intersection[c, r] (* A005836 *)

Formula

Conjecture: a(n) = 3*A003278(n) - 2 = (A055246(n) + 1)/2. - L. Edson Jeffery, Nov 25 2015

Conjecture: a(n) = A190640(n)/2. - Michel Marcus, Aug 24 2016

Conjecture: a(n) = A003278(2n-1). - Arie Bos, Aug 07 2022