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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.

%I A191203 #8 Apr 27 2014 00:18:43
%S A191203 1,2,4,5,8,10,16,17,20,26,32,34,40,52,64,65,68,80,101,104,128,130,136,
%T A191203 160,202,208,256,257,260,272,290,320,401,404,416,512,514,520,544,580,
%U A191203 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
%N 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.
%C A191203 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.
%C A191203 Examples include A190803, A191106, A191113, and these:
%C A191203   A191203: 2x, 1+x^2
%C A191203   A191211: 1+2x, 1+x^2
%C A191203   A191281: 2x, x^2-x+1
%C A191203   A191282: 2x, x^2+x+1
%C A191203   A191283: 2x, x(x+1)/2
%C A191203   A191284: floor(3x/2), 2x
%C A191203   A191285: 3x, floor((x^2)/2)
%C A191203   A191286: 3x, 1+x^2
%C A191203   A191287: floor(3x/2), 3x
%C A191203   A191288: 2x, floor((x^2)/3)
%C A191203   A191289: 3x-1, x^2
%C A191203   A191290: 2x+1, x(x+1)/2
%C A191203 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 -> ....
%H A191203 Reinhard Zumkeller, <a href="/A191203/b191203.txt">Table of n, a(n) for n = 1..10000</a>
%e A191203 1 -> 2 -> 4,5 -> 8,10,17,26 ->
%t A191203 g = 12; Union[Flatten[NestList[{2 #, 1 + #^2} &, 1, g]]]
%t A191203 (*  A191203; use g>11 to get all terms up to 4096 *)
%o A191203 (Haskell)
%o A191203 import Data.Set (singleton, deleteFindMin, insert)
%o A191203 a191203 n = a191203_list !! (n-1)
%o A191203 a191203_list = f $ singleton 1 where
%o A191203    f s = m : f (insert (2 * m) $ insert (m ^ 2 + 1) s')
%o A191203          where (m, s') = deleteFindMin s
%o A191203 -- _Reinhard Zumkeller_, Apr 18 2014
%Y A191203 Cf. A190803, A191106, A191113.
%K A191203 nonn
%O A191203 1,2
%A A191203 _Clark Kimberling_, May 29 2011