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A191113
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Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 4x-2 are in a.
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81
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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
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OFFSET
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1,2
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COMMENTS
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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:
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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).
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LINKS
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FORMULA
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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.
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EXAMPLE
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1 -> 2 -> 4,6 -> 10,14,16,22 ->
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MAPLE
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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:
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MATHEMATICA
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h = 3; i = -2; j = 4; k = -2; f = 1; g = 8;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]
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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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