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

1,2

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, 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.

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.

Example

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

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. *)

Prog

(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

Crossrefs

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

Sequence in context: A251599 A270151 A009388 * A125871 A141399 A104737

Adjacent sequences: A190800 A190801 A190802 * A190804 A190805 A190806

Keyword

nonn

Author

Clark Kimberling, May 25 2011

Extensions

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

Status

approved