A190803 - OEIS

%I #26 Nov 29 2015 13:46:27

%S 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 81,86,87,89,98,99,101,105,113,122,129,131,134,135,149,152,153,158,

%U 159,161,170,171,173,177,194,195,197,201,203,209,225,230,239,242

%N 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 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 Examples, where [A......] indicates a conjecture:

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

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

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

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

%C ...

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

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

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

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

%C ...

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

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

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

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

%C ...

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

%H Reinhard Zumkeller, <a href="/A190803/b190803.txt">Table of n, a(n) for n = 1..10000</a>

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

%t h = 2; i = -1; j = 3; k = -1; f = 1; g = 10;

%t a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190803 *)

%t b = (a + 1)/2; c = (a + 1)/3; r = Range[1, 300];

%t d = Intersection[b, r] (* A190841 *)

%t e = Intersection[c, r] (* A190842 *)

%t (* 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 (Haskell)

%o import Data.Set (singleton, deleteFindMin, insert)

%o a190803 n = a190803_list !! (n-1)

%o a190803_list = 1 : f (singleton 2)

%o    where f s = m : (f $ insert (2*m-1) $ insert (3*m-1) s')

%o              where (m, s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, Jun 01 2011

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

%K nonn

%O 1,2

%A _Clark Kimberling_, May 25 2011

%E a(34)=225 inserted by _Reinhard Zumkeller_, Jun 01 2011