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A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

%I A190803 #26 Nov 29 2015 13:46:27
%S A190803 1,2,3,5,8,9,14,15,17,23,26,27,29,33,41,44,45,50,51,53,57,65,68,77,80,
%T A190803 81,86,87,89,98,99,101,105,113,122,129,131,134,135,149,152,153,158,
%U A190803 159,161,170,171,173,177,194,195,197,201,203,209,225,230,239,242
%N A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.
%C A190803 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.
%C A190803 Examples, where [A......] indicates a conjecture:
%C A190803 A190803: (h,i,j,k)=(2,-1,3,-1); d=A190841, e=A190842
%C A190803 A190804: (h,i,j,k)=(2,-1,3,0); d=[A190803], e=A190844
%C A190803 A190805: (h,i,j,k)=(2,-1,3,1); d=A190845, e=[A190808]
%C A190803 A190806: (h,i,j,k)=(2,-1,3,2); d=[A190804], e=A190848
%C A190803 ...
%C A190803 A190807: (h,i,j,k)=(2,0,3,-1); d=A190849, e=A190850
%C A190803 A003586: (h,i,j,k)=(2,0,3,0); d=e=A003586
%C A190803 A190808: (h,i,j,k)=(2,0,3,1); d=A190851, e=A190852
%C A190803 A190809: (h,i,j,k)=(2,0,3,2); d=A190853, e=A190854
%C A190803 ...
%C A190803 A190810: (h,i,j,k)=(2,1,3,-1); d=A190855, e=A190856
%C A190803 A190811: (h,i,j,k)=(2,1,3,0); d=A002977, e=A190857
%C A190803 A002977: (h,i,j,k)=(2,1,3,1); d=A190858, e=A190859
%C A190803 A190812: (h,i,j,k)=(2,1,3,2); d=A069353, e=[A190812]
%C A190803 ...
%C A190803 For h=j=3, see A191106; for h=3 and j=4, see A191113.
%H A190803 Reinhard Zumkeller, <a href="/A190803/b190803.txt">Table of n, a(n) for n = 1..10000</a>
%H A190803 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, 10 (2007) 1-13.
%e A190803 1 -> 2 -> 3,5 -> 8,9,14 -> 15,17,23,26,27,41 -> ...
%t A190803 h = 2; i = -1; j = 3; k = -1; f = 1; g = 10;
%t A190803 a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190803 *)
%t A190803 b = (a + 1)/2; c = (a + 1)/3; r = Range[1, 300];
%t A190803 d = Intersection[b, r] (* A190841 *)
%t A190803 e = Intersection[c, r] (* A190842 *)
%t A190803 (* Regarding this program - useful for many choices of h,i,j,k,f,g - the depth g must be chosen with care - that is, large enough.  Assuming that h<=j, the least new terms in successive nests a are typically iterates of hx+i, starting from x=1.  If, for example, h=2 and i=0, the least terms are 2,4,8,...,2^g, so that g>=9 ensures inclusion of all the desired terms <=256. *)
%o A190803 (Haskell)
%o A190803 import Data.Set (singleton, deleteFindMin, insert)
%o A190803 a190803 n = a190803_list !! (n-1)
%o A190803 a190803_list = 1 : f (singleton 2)
%o A190803    where f s = m : (f $ insert (2*m-1) $ insert (3*m-1) s')
%o A190803              where (m, s') = deleteFindMin s
%o A190803 -- _Reinhard Zumkeller_, Jun 01 2011
%Y A190803 Cf. A002977, A003586, A190804-A190812, A190841-A190860.
%K A190803 nonn
%O A190803 1,2
%A A190803 _Clark Kimberling_, May 25 2011
%E A190803 a(34)=225 inserted by _Reinhard Zumkeller_, Jun 01 2011