A191113 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 4x-2 are in a.

1, 2, 4, 6, 10, 14, 16, 22, 28, 38, 40, 46, 54, 62, 64, 82, 86, 110, 112, 118, 136, 150, 158, 160, 182, 184, 190, 214, 244, 246, 254, 256, 326, 328, 334, 342, 352, 406, 438, 446, 448, 470, 472, 478, 542, 544, 550, 568, 598, 630, 638, 640, 726, 730, 734, 736, 758, 760, 766, 854, 974, 976, 982, 1000, 1014, 1022, 1024, 1054, 1216

Offset: 1

Author

Clark Kimberling, May 27 2011

Keywords

• nonn

Comments

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then hx+i and jx+k are in a, where h,i,j,k are integers." If m>1, at least one of the numbers b(m)=(a(m)-i)/h and c(m)=(a(m)-k)/j is in the set N of natural numbers. Let d(n) be the n-th b(m) in N, and let e(n) be the n-th c(m) in N. Note that a is a subsequence of both d and e. Examples:

A191113: (h,i,j,k)=(3,-2,4,-2); d=A191146, e=A191149

A191114: (h,i,j,k)=(3,-2,4,-1); d=A191151, e=A191121

A191115: (h,i,j,k)=(3,-2,4,0); d=A191113, e=A191154

A191116: (h,i,j,k)=(3,-2,4,1); d=A191155 e=A191129

A191117: (h,i,j,k)=(3,-2,4,2); d=A191157, e=A191158

A191118: (h,i,j,k)=(3,-2,4,3); d=A191114, e=A191138

...

A191119: (h,i,j,k)=(3,-1,4,-3); d=A191120, e=A191163

A191120: (h,i,j,k)=(3,-1,4,-2); d=A191129, e=A191165

A191121: (h,i,j,k)=(3,-1,4,-1); d=A191166, e=A191167

A191122: (h,i,j,k)=(3,-1,4,0); d=A191168, e=A191169

A191123: (h,i,j,k)=(3,-1,4,1); d=A191170, e=A191171

A191124: (h,i,j,k)=(3,-1,4,2); d=A191172, e=A191173

A191125: (h,i,j,k)=(3,-1,4,3); d=A191174, e=A191175

...

A191126: (h,i,j,k)=(3,0,4,-3); d=A191128, e=A191177

A191127: (h,i,j,k)=(3,0,4,-2); d=A191178, e=A191179

A191128: (h,i,j,k)=(3,0,4,-1); d=A191180, e=A191181

A025613: (h,i,j,k)=(3,0,4,0); d=e=A025613

A191129: (h,i,j,k)=(3,0,4,1); d=A191182, e=A191183

A191130: (h,i,j,k)=(3,0,4,2); d=A191184, e=A191185

A191131: (h,i,j,k)=(3,0,4,3); d=A191186, e=A191187

...

A191132: (h,i,j,k)=(3,1,4,-3); d=A191135, e=A191189

A191133: (h,i,j,k)=(3,1,4,-2); d=A191190, e=A191191

A191134: (h,i,j,k)=(3,1,4,-1); d=A191192, e=A191193

A191135: (h,i,j,k)=(3,1,4,0); d=A191136, e=A191195

A191136: (h,i,j,k)=(3,1,4,1); d=A191196, e=A191197

A191137: (h,i,j,k)=(3,1,4,2); d=A191198, e=A191199

A191138: (h,i,j,k)=(3,1,4,3); d=A191200, e=A191201

...

A191139: (h,i,j,k)=(3,2,4,-3); d=A191143, e=A191119

A191140: (h,i,j,k)=(3,2,4,-2); d=A191204, e=A191205

A191141: (h,i,j,k)=(3,2,4,-1); d=A191206, e=A191207

A191142: (h,i,j,k)=(3,2,4,0); d=A191208, e=A191209

A191143: (h,i,j,k)=(3,2,4,1); d=A191210, e=A191136

A191144: (h,i,j,k)=(3,2,4,2); d=A191212, e=A191213

A191145: (h,i,j,k)=(3,2,4,3); d=e=A191145

...

Representative divisibility properties:

if s=A191116, then 2|(s+1), 4|(s+3), and 8|(s+3) for n>1; if s=A191117, then 10|(s+4) for n>1.

For lists of other "rules sequences" see A190803 (h=2 and j=3) and A191106 (h=j=3).

Examples

			1 -> 2 -> 4,6 -> 10,14,16,22 ->
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 1-13.

Crossrefs

Cf. A190803, A191106.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a191113 n = a191113_list !! (n-1)
a191113_list = 1 : f (singleton 2)
   where f s = m : (f $ insert (3*m-2) $ insert (4*m-2) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

Maple

N:= 2000: # to get all terms <= N
S:= {}: agenda:= {1}:
while nops(agenda) > 0 do
  S:= S union agenda;
  agenda:= select(`<=`,map(t -> (3*t-2,4*t-2),agenda) minus S, N)
od:
sort(convert(S,list)); # Robert Israel, Dec 22 2015

Mathematica

h = 3; i = -2; j = 4; k = -2; f = 1; g = 8;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]
(* a=A191113; regarding g, see the Mathematica note at A190803 *)
b = (a + 2)/3; c = (a + 2)/4; r = Range[1, 900];
d = Intersection[b, r] (* A191146 *)
e = Intersection[c, r] (* A191149 *)
m = a/2 (* divisibility property *)

Formula

a(1)=1, and if x is in a then 3x-2 and 4x-2 are in a; the terms of a are listed in without repetitions, in increasing order.