A167198 Fractal sequence of the interspersion A083047.

1, 1, 2, 1, 2, 3, 1, 4, 2, 3, 5, 1, 4, 6, 2, 7, 3, 5, 8, 1, 9, 4, 6, 10, 2, 7, 11, 3, 12, 5, 8, 13, 1, 9, 14, 4, 15, 6, 10, 16, 2, 17, 7, 11, 18, 3, 12, 19, 5, 20, 8, 13, 21, 1, 22, 9, 14, 23, 4, 15, 24, 6, 25, 10, 16, 26, 2, 17, 27, 7, 28, 11, 18, 29, 3, 30, 12, 19, 31, 5, 20, 32, 8, 33, 13

Offset: 1

Author

Clark Kimberling, Oct 30 2009

Keywords

• nonn

Comments

As a fractal sequence, if the first occurrence of each term is deleted, the remaining sequence is the original. In general, the interspersion of a fractal sequence is constructed by rows: row r consists of all n, such that a(n)=r; in particular, A083047 is constructed in this way from A167198.

a(n-1) gives the row number which contains n in the dual Wythoff array A126714 (beginning the row count at 1), see also A223025 and A019586. - Casey Mongoven, Mar 11 2013

Examples

			To produce row 5, first write row 4: 2,3,1, then place 4 right before 2, and then place 5 right before 1, getting 4,2,3,5,1.
		

References

• Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

Links

• Clark Kimberling, The first column of an interspersion, The Fibonacci Quarterly 32 (1994), 301-315.

Crossrefs

Cf. A003603, A083047, A035513, A000045.

Formula

Following is a construction that avoids reference to A083047.

Write initial rows:

Row 1: .... 1

Row 2: .... 1

Row 3: .... 2..1

Row 4: .... 2..3..1

For n>=4, to form row n+1, let k be the least positive integer not yet used; write row n, and right before the first number that is also in row n-1, place k; right before the next number that is also in row n-1, place k+1, and continue. A167198 is the concatenation of the rows. (If "before" is replaced by "after", the resulting fractal sequence is A003603, and the associated interspersion is the Wythoff array, A035513.)