0,2

Comments from Benoit CLOITRE, Mar 09 2007: (Start) The sequence can be construct as follows using parentheses (NP means "term not in parentheses"):

Start from the natural integers:

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,...

Step 1 : put the least NP "1" in parentheses and every 2 terms giving:

(1),2,(3),4,(5),6,(7),8,(9),10,(11),12,(13),14,(15),16,(17),18,(19),...

Step 2 : put the least NP "2" in 2 parentheses and every 3 NP giving:

(1),((2)),(3),4,(5),6,(7),((8)),(9),10,(11),12,(13),((14)),(15),16,(17),...

So that between 2 consecutives ((x)) there are 2 NP.

Step 3 : put the least NP "4" in 3 parentheses and every 4 NP giving:

(1),((2)),(3),(((4))),(5),6,(7),((8)),(9),10,(11),12,(13),((14)),(15),(((16))),...

So that between 2 consecutives (((x))) there are 3 NP.

Step 4 : put the least NP "6" in 4 parentheses and every 5 NP giving:

(1),((2)),(3),(((4))),(5),((((6)))),(7),((8)),(9),10,(11),12,(13),((14)),(15),(((16))),...

So that between 2 consecutives ((((x)))) there are 4 NP.

Iterating the process indefinitely yields:

(1),((2)),(3),(((4))),(5),((((6)))),(7),((8)),(9),(((((10))))),(11),...

Count the parentheses:

1,2,1,3,1,4,1,2,1,5,1,... - this is the sequence. (End)

Comment from Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 26 2007: "A simpler way to construct the sequence : start from:

1,_,1,_,1,_,1,_,1,_,1,_,1,_,1,... where 1's are spaced by one hole

fill first hole with 2 and leave 2 holes between two 2's giving:

1,2,1,_,1,_,1,2,1,_,1,_,1,2,1,...

fill new first hole with 3 and leave 3 holes between two 3's giving:

1,2,1,3,1,_,1,2,1,_,1,_,1,2,1,3...

iterating the process indefinitely yields the sequence."

Ordinal transform of A130747 - Benoit Cloitre, Aug 03 2007

Although A028920 and A130747 are not fractal sequences (according to Kimberling's definition) we say they are "mutual fractal sequences" since the ordinal transform of one gives the other. - Benoit Cloitre, Aug 03 2007

L. K. Mitchell, Table of n, a(n) for n=0..3820

D. M. Broline and D. E. Loeb (daniel.loeb(AT)verizon.net), The combinatorics of Mancala-Type games: Ayo, Tchoukaillon and 1/Pi, J. Undergrad. Math. Applic., vol. 16 (1995), pp. 21-36.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Index entries for sequences generated by sieves

Franklin T. Adams-Watters, Doubly Fractal Sequences and ordinal transform

a(2n-1)=1, a(2n)=1+A104706(n). - Benoit CLOITRE, Mar 09 2007

The sieve of A007952 processes n in the a(n)-th pass. a(A007952(n)) = n+1.

Cf. A002491, A007952, A028920, A028931, A028932, A028933.

Cf. A130747.

Sequence in context: A102547 A087267 A128267 this_sequence A055396 A057499 A064839

Adjacent sequences: A028917 A028918 A028919 this_sequence A028921 A028922 A028923

nonn

David W. Wilson (davidwwilson(AT)comcast.net)