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 A191113 Increasing sequence generated by these rules:  a(1)=1, and if x is in a then 3x-2 and 4x-2 are in a. 82
 1, 2, 4, 6, 10, 14, 16, 22, 28, 38, 40, 46, 54, 62, 64, 82, 86, 110, 112, 118, 136, 150, 158, 160, 182, 184, 190, 214, 244, 246, 254, 256, 326, 328, 334, 342, 352, 406, 438, 446, 448, 470, 472, 478, 542, 544, 550, 568, 598, 630, 638, 640, 726, 730, 734, 736, 758, 760, 766, 854, 974, 976, 982, 1000, 1014, 1022, 1024, 1054, 1216 (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: A191113: (h,i,j,k)=(3,-2,4,-2); d=A191146, e=A191149 A191114: (h,i,j,k)=(3,-2,4,-1); d=A191151, e=A191121 A191115: (h,i,j,k)=(3,-2,4,0); d=A191113, e=A191154 A191116: (h,i,j,k)=(3,-2,4,1); d=A191155 e=A191129 A191117: (h,i,j,k)=(3,-2,4,2); d=A191157, e=A191158 A191118: (h,i,j,k)=(3,-2,4,3); d=A191114, e=A191138 ... A191119: (h,i,j,k)=(3,-1,4,-3); d=A191120, e=A191163 A191120: (h,i,j,k)=(3,-1,4,-2); d=A191129, e=A191165 A191121: (h,i,j,k)=(3,-1,4,-1); d=A191166, e=A191167 A191122: (h,i,j,k)=(3,-1,4,0); d=A191168, e=A191169 A191123: (h,i,j,k)=(3,-1,4,1); d=A191170, e=A191171 A191124: (h,i,j,k)=(3,-1,4,2); d=A191172, e=A191173 A191125: (h,i,j,k)=(3,-1,4,3); d=A191174, e=A191175 ... A191126: (h,i,j,k)=(3,0,4,-3); d=A191128, e=A191177 A191127: (h,i,j,k)=(3,0,4,-2); d=A191178, e=A191179 A191128: (h,i,j,k)=(3,0,4,-1); d=A191180, e=A191181 A025613: (h,i,j,k)=(3,0,4,0); d=e=A025613 A191129: (h,i,j,k)=(3,0,4,1); d=A191182, e=A191183 A191130: (h,i,j,k)=(3,0,4,2); d=A191184, e=A191185 A191131: (h,i,j,k)=(3,0,4,3); d=A191186, e=A191187 ... A191132: (h,i,j,k)=(3,1,4,-3); d=A191135, e=A191189 A191133: (h,i,j,k)=(3,1,4,-2); d=A191190, e=A191191 A191134: (h,i,j,k)=(3,1,4,-1); d=A191192, e=A191193 A191135: (h,i,j,k)=(3,1,4,0); d=A191136, e=A191195 A191136: (h,i,j,k)=(3,1,4,1); d=A191196, e=A191197 A191137: (h,i,j,k)=(3,1,4,2); d=A191198, e=A191199 A191138: (h,i,j,k)=(3,1,4,3); d=A191200, e=A191201 ... A191139: (h,i,j,k)=(3,2,4,-3); d=A191143, e=A191119 A191140: (h,i,j,k)=(3,2,4,-2); d=A191204, e=A191205 A191141: (h,i,j,k)=(3,2,4,-1); d=A191206, e=A191207 A191142: (h,i,j,k)=(3,2,4,0); d=A191208, e=A191209 A191143: (h,i,j,k)=(3,2,4,1); d=A191210, e=A191136 A191144: (h,i,j,k)=(3,2,4,2); d=A191212, e=A191213 A191145: (h,i,j,k)=(3,2,4,3); d=e=A191145 ... Representative divisibility properties: if s=A191116, then 2|(s+1), 4|(s+3), and 8|(s+3) for n>1; if s=A191117, then 10|(s+4) for n>1. For lists of other "rules sequences" see A190803 (h=2 and j=3) and A191106 (h=j=3). 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, Article 07.1.5, 10 (2007) 1-13. FORMULA a(1)=1, and if x is in a then 3x-2 and 4x-2 are in a; the terms of a are listed in without repetitions, in increasing order. EXAMPLE 1 -> 2 -> 4,6 -> 10,14,16,22 -> MAPLE N:= 2000: # to get all terms <= N S:= {}: agenda:= {1}: while nops(agenda) > 0 do   S:= S union agenda;   agenda:= select(`<=`, map(t -> (3*t-2, 4*t-2), agenda) minus S, N) od: sort(convert(S, list)); # Robert Israel, Dec 22 2015 MATHEMATICA h = 3; i = -2; j = 4; k = -2; f = 1; g = 8; a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]] (* a=A191113; regarding g, see the Mathematica note at A190803 *) b = (a + 2)/3; c = (a + 2)/4; r = Range[1, 900]; d = Intersection[b, r] (* A191146 *) e = Intersection[c, r] (* A191149 *) m = a/2 (* divisibility property *) PROG (Haskell) import Data.Set (singleton, deleteFindMin, insert) a191113 n = a191113_list !! (n-1) a191113_list = 1 : f (singleton 2)    where f s = m : (f \$ insert (3*m-2) \$ insert (4*m-2) s')              where (m, s') = deleteFindMin s -- Reinhard Zumkeller, Jun 01 2011 CROSSREFS Cf. A190803, A191106. Sequence in context: A005574 A109807 A259645 * A125964 A288526 A139544 Adjacent sequences:  A191110 A191111 A191112 * A191114 A191115 A191116 KEYWORD nonn AUTHOR Clark Kimberling, May 27 2011 STATUS approved

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Last modified August 18 14:30 EDT 2018. Contains 313832 sequences. (Running on oeis4.)