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

1,2

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

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

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

Cf. A190803, A191106, A191113.

Sequence in context: A126684 A089653 A180252 * A216686 A114652 A191288

Adjacent sequences: A191200 A191201 A191202 * A191204 A191205 A191206

Keyword

nonn

Author

Clark Kimberling, May 29 2011

Status

approved