A191178 -id:A191178

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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.

+10
81

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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).

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.

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.

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A190803, A191106.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 27 2011

STATUS

approved

A191127

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

+10
4

1, 2, 3, 6, 9, 10, 18, 22, 27, 30, 34, 38, 54, 66, 70, 81, 86, 90, 102, 106, 114, 118, 134, 150, 162, 198, 210, 214, 243, 258, 262, 270, 278, 306, 318, 322, 342, 354, 358, 402, 406, 422, 450, 454, 470, 486, 534, 594, 598, 630, 642, 646, 729, 774, 786, 790, 810, 834, 838, 854, 918, 954, 966, 970, 1026, 1030, 1046, 1062, 1074, 1078

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

See A191113.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

MATHEMATICA

h = 3; i = 0; j = 4; k = -2; f = 1; g = 9;

a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]] (* A191127 *)

b = a/3; c = (a + 2)/4; r = Range[1, 1500];

d = Intersection[b, r] (* A191178 *)

e = Intersection[c, r] (* A191179 *)

Nest[Flatten[{#, 3#, 4#-2}]&, 1, 7]//Union (* Harvey P. Dale, Jul 20 2021 *)