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.

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, 81, 86, 87, 89, 98, 99, 101, 105, 113, 122, 129, 131, 134, 135, 149, 152, 153, 158, 159, 161, 170, 171, 173, 177, 194, 195, 197, 201, 203, 209, 225, 230, 239, 242

Offset: 1

Author

Clark Kimberling, May 25 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, where [A......] indicates a conjecture:

A190803: (h,i,j,k)=(2,-1,3,-1); d=A190841, e=A190842

A190804: (h,i,j,k)=(2,-1,3,0); d=[A190803], e=A190844

A190805: (h,i,j,k)=(2,-1,3,1); d=A190845, e=[A190808]

A190806: (h,i,j,k)=(2,-1,3,2); d=[A190804], e=A190848

...

A190807: (h,i,j,k)=(2,0,3,-1); d=A190849, e=A190850

A003586: (h,i,j,k)=(2,0,3,0); d=e=A003586

A190808: (h,i,j,k)=(2,0,3,1); d=A190851, e=A190852

A190809: (h,i,j,k)=(2,0,3,2); d=A190853, e=A190854

...

A190810: (h,i,j,k)=(2,1,3,-1); d=A190855, e=A190856

A190811: (h,i,j,k)=(2,1,3,0); d=A002977, e=A190857

A002977: (h,i,j,k)=(2,1,3,1); d=A190858, e=A190859

A190812: (h,i,j,k)=(2,1,3,2); d=A069353, e=[A190812]

...

For h=j=3, see A191106; for h=3 and j=4, see A191113.

Examples

			1 -> 2 -> 3,5 -> 8,9,14 -> 15,17,23,26,27,41 -> ...
		

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, 10 (2007) 1-13.

Crossrefs

Cf. A002977, A003586, A190804-A190812, A190841-A190860.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a190803 n = a190803_list !! (n-1)
a190803_list = 1 : f (singleton 2)
   where f s = m : (f $ insert (2*m-1) $ insert (3*m-1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

Mathematica

h = 2; i = -1; j = 3; k = -1; f = 1; g = 10;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190803 *)
b = (a + 1)/2; c = (a + 1)/3; r = Range[1, 300];
d = Intersection[b, r] (* A190841 *)
e = Intersection[c, r] (* A190842 *)
(* 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. *)

Extensions

a(34)=225 inserted by Reinhard Zumkeller, Jun 01 2011