A135287 a(0)=1; for n > 0, a(n) = a(n-1)+n if a(n-1) is odd, else a(n) = a(n-1)/2.

1, 2, 1, 4, 2, 1, 7, 14, 7, 16, 8, 4, 2, 1, 15, 30, 15, 32, 16, 8, 4, 2, 1, 24, 12, 6, 3, 30, 15, 44, 22, 11, 43, 76, 38, 19, 55, 92, 46, 23, 63, 104, 52, 26, 13, 58, 29, 76, 38, 19, 69, 120, 60, 30, 15, 70, 35, 92, 46, 23, 83

Offset: 0

Author

Ctibor O. Zizka, Dec 03 2007, Dec 05 2007

Keywords

• nonn
• easy

Comments

Let a(0), C1, C2, C be integers. Consider the sequence a(n) = a(n-1) + C1*n + C2 if a(n-1) is not divisible by C or a(n) = a(n-1)/C otherwise.

For a fixed C1, C2, C this sequence shows chaotic behavior for some a(0) and a highly regular behavior for other a(0).

The parameter C1 tells how many regular subclasses are there.

The sequence grows roughly as a(n) ~ n*const.

Here C = 2. Other sequences showing very interesting behavior have C = power of 2.

Example: C1=3, C2=10, C=3. Thus a(n)= a(n-1)+3*n+10 if a(n-1) is not divisible by 3, or a(n)= a(n-1)/3 otherwise. There are 2 classes:

a regular class with 3 subclasses (C1=3) for initial values

{a(0)=3,38,79,...}

{a(0)=1,8,12,42,47,49,63,77,88,...}

{a(0)=2,43,45,...}

and a "chaotic" class for other initial values a(0).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A008336, A005132, A135294.

Cf. A090895.

Programs

Haskell

a135287 n = a135287_list !! n
a135287_list = 1 : f 1 1 where
   f x y = z : f (x + 1) z where
        z = if m == 0 then y' else x + y; (y',m) = divMod y 2
-- Reinhard Zumkeller, Mar 02 2012

Maple

A135287 := proc(n) option remember ; if n = 0 then 1 ; elif A135287(n-1) mod 2 = 0 then A135287(n-1)/2 ; else n+A135287(n-1) ; fi ; end: seq(A135287(n),n=0..60) ; # R. J. Mathar, Dec 12 2007

Mathematica

nxt[{n_,a_}]:={n+1,If[OddQ[a],a+n+1,a/2]}; NestList[nxt,{0,1},60][[;;,2]] (* Harvey P. Dale, Mar 02 2023 *)

Extensions

More terms from R. J. Mathar, Dec 12 2007

Offset fixed by Reinhard Zumkeller, Mar 02 2012