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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.

%I A191113 #22 Dec 23 2015 02:53:46
%S A191113 1,2,4,6,10,14,16,22,28,38,40,46,54,62,64,82,86,110,112,118,136,150,
%T A191113 158,160,182,184,190,214,244,246,254,256,326,328,334,342,352,406,438,
%U A191113 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
%N 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.
%C A191113 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:
%C A191113 A191113: (h,i,j,k)=(3,-2,4,-2); d=A191146, e=A191149
%C A191113 A191114: (h,i,j,k)=(3,-2,4,-1); d=A191151, e=A191121
%C A191113 A191115: (h,i,j,k)=(3,-2,4,0); d=A191113, e=A191154
%C A191113 A191116: (h,i,j,k)=(3,-2,4,1); d=A191155 e=A191129
%C A191113 A191117: (h,i,j,k)=(3,-2,4,2); d=A191157, e=A191158
%C A191113 A191118: (h,i,j,k)=(3,-2,4,3); d=A191114, e=A191138
%C A191113 ...
%C A191113 A191119: (h,i,j,k)=(3,-1,4,-3); d=A191120, e=A191163
%C A191113 A191120: (h,i,j,k)=(3,-1,4,-2); d=A191129, e=A191165
%C A191113 A191121: (h,i,j,k)=(3,-1,4,-1); d=A191166, e=A191167
%C A191113 A191122: (h,i,j,k)=(3,-1,4,0); d=A191168, e=A191169
%C A191113 A191123: (h,i,j,k)=(3,-1,4,1); d=A191170, e=A191171
%C A191113 A191124: (h,i,j,k)=(3,-1,4,2); d=A191172, e=A191173
%C A191113 A191125: (h,i,j,k)=(3,-1,4,3); d=A191174, e=A191175
%C A191113 ...
%C A191113 A191126: (h,i,j,k)=(3,0,4,-3); d=A191128, e=A191177
%C A191113 A191127: (h,i,j,k)=(3,0,4,-2); d=A191178, e=A191179
%C A191113 A191128: (h,i,j,k)=(3,0,4,-1); d=A191180, e=A191181
%C A191113 A025613: (h,i,j,k)=(3,0,4,0); d=e=A025613
%C A191113 A191129: (h,i,j,k)=(3,0,4,1); d=A191182, e=A191183
%C A191113 A191130: (h,i,j,k)=(3,0,4,2); d=A191184, e=A191185
%C A191113 A191131: (h,i,j,k)=(3,0,4,3); d=A191186, e=A191187
%C A191113 ...
%C A191113 A191132: (h,i,j,k)=(3,1,4,-3); d=A191135, e=A191189
%C A191113 A191133: (h,i,j,k)=(3,1,4,-2); d=A191190, e=A191191
%C A191113 A191134: (h,i,j,k)=(3,1,4,-1); d=A191192, e=A191193
%C A191113 A191135: (h,i,j,k)=(3,1,4,0); d=A191136, e=A191195
%C A191113 A191136: (h,i,j,k)=(3,1,4,1); d=A191196, e=A191197
%C A191113 A191137: (h,i,j,k)=(3,1,4,2); d=A191198, e=A191199
%C A191113 A191138: (h,i,j,k)=(3,1,4,3); d=A191200, e=A191201
%C A191113 ...
%C A191113 A191139: (h,i,j,k)=(3,2,4,-3); d=A191143, e=A191119
%C A191113 A191140: (h,i,j,k)=(3,2,4,-2); d=A191204, e=A191205
%C A191113 A191141: (h,i,j,k)=(3,2,4,-1); d=A191206, e=A191207
%C A191113 A191142: (h,i,j,k)=(3,2,4,0); d=A191208, e=A191209
%C A191113 A191143: (h,i,j,k)=(3,2,4,1); d=A191210, e=A191136
%C A191113 A191144: (h,i,j,k)=(3,2,4,2); d=A191212, e=A191213
%C A191113 A191145: (h,i,j,k)=(3,2,4,3); d=e=A191145
%C A191113 ...
%C A191113 Representative divisibility properties:
%C A191113 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.
%C A191113 For lists of other "rules sequences" see A190803 (h=2 and j=3) and A191106 (h=j=3).
%H A191113 Reinhard Zumkeller, <a href="/A191113/b191113.txt">Table of n, a(n) for n = 1..10000</a>
%H A191113 David Garth and Adam Gouge, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Garth/garth14.html">Affinely Self-Generating Sets and Morphisms</a>, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 1-13.
%F A191113 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.
%e A191113 1 -> 2 -> 4,6 -> 10,14,16,22 ->
%p A191113 N:= 2000: # to get all terms <= N
%p A191113 S:= {}: agenda:= {1}:
%p A191113 while nops(agenda) > 0 do
%p A191113   S:= S union agenda;
%p A191113   agenda:= select(`<=`,map(t -> (3*t-2,4*t-2),agenda) minus S, N)
%p A191113 od:
%p A191113 sort(convert(S,list)); # _Robert Israel_, Dec 22 2015
%t A191113 h = 3; i = -2; j = 4; k = -2; f = 1; g = 8;
%t A191113 a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]
%t A191113 (* a=A191113; regarding g, see the Mathematica note at A190803 *)
%t A191113 b = (a + 2)/3; c = (a + 2)/4; r = Range[1, 900];
%t A191113 d = Intersection[b, r] (* A191146 *)
%t A191113 e = Intersection[c, r] (* A191149 *)
%t A191113 m = a/2 (* divisibility property *)
%o A191113 (Haskell)
%o A191113 import Data.Set (singleton, deleteFindMin, insert)
%o A191113 a191113 n = a191113_list !! (n-1)
%o A191113 a191113_list = 1 : f (singleton 2)
%o A191113    where f s = m : (f $ insert (3*m-2) $ insert (4*m-2) s')
%o A191113              where (m, s') = deleteFindMin s
%o A191113 -- _Reinhard Zumkeller_, Jun 01 2011
%Y A191113 Cf. A190803, A191106.
%K A191113 nonn
%O A191113 1,2
%A A191113 _Clark Kimberling_, May 27 2011