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A191203
Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x and 1+x^2 are in a.
13
1, 2, 4, 5, 8, 10, 16, 17, 20, 26, 32, 34, 40, 52, 64, 65, 68, 80, 101, 104, 128, 130, 136, 160, 202, 208, 256, 257, 260, 272, 290, 320, 401, 404, 416, 512, 514, 520, 544, 580, 640, 677, 802, 808, 832, 1024, 1025, 1028, 1040, 1088, 1157, 1160, 1280, 1354, 1601, 1604, 1616, 1664, 2048, 2050, 2056, 2080, 2176, 2314, 2320, 2560
OFFSET
1,2
COMMENTS
The method generalizes: a finite set F={f} of functions f:N->N and finite set G of numbers generate a set S by these rules: (1) every element of G is in S, and (2) if x is in S then f(x) is in S for every f in F. The sequence a results by taking the numbers in S in increasing order.
Examples include A190803, A191106, A191113, and these:
A191203: 2x, 1+x^2
A191211: 1+2x, 1+x^2
A191281: 2x, x^2-x+1
A191282: 2x, x^2+x+1
A191283: 2x, x(x+1)/2
A191284: floor(3x/2), 2x
A191285: 3x, floor((x^2)/2)
A191286: 3x, 1+x^2
A191287: floor(3x/2), 3x
A191288: 2x, floor((x^2)/3)
A191289: 3x-1, x^2
A191290: 2x+1, x(x+1)/2
For A191203 and other such sequences, the depth g for the NestList in the Mathematica program must be large enough to generate as many terms as required by the user. For example, the rules 2x and 1+x^2, starting with x=1, successively generate set of numbers whose minima are powers of 2: 1->2->4-> ... 2^g -> ....
LINKS
EXAMPLE
1 -> 2 -> 4,5 -> 8,10,17,26 ->
MATHEMATICA
g = 12; Union[Flatten[NestList[{2 #, 1 + #^2} &, 1, g]]]
(* A191203; use g>11 to get all terms up to 4096 *)
PROG
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a191203 n = a191203_list !! (n-1)
a191203_list = f $ singleton 1 where
f s = m : f (insert (2 * m) $ insert (m ^ 2 + 1) s')
where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 18 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 29 2011
STATUS
approved