A191115 -id:A191115 - cp's OEIS frontend

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

Clark Kimberling, May 27 2011

nonn

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).

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

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.

Cf. A190803, A191106.

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

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

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 *)

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.

cp's OEIS Frontend

A191154 Integers in (A191115)/4; contains A191115 as a proper subsequence.

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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.

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