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.

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

Author

Clark Kimberling, May 29 2011

Keywords

• nonn

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 -> ....

Examples

			1 -> 2 -> 4,5 -> 8,10,17,26 ->
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A190803, A191106, A191113.

Programs

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

Mathematica

g = 12; Union[Flatten[NestList[{2 #, 1 + #^2} &, 1, g]]]
(*  A191203; use g>11 to get all terms up to 4096 *)