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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. 32
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 22 21:38 EDT 2024. Contains 372758 sequences. (Running on oeis4.)