A174736 Number of nodes with index 2^2k in a dyadic tree build alternatively with the schema "l" between the index 2^2k and 2^(2k+1) - 1, and the schema "^" between the index 2^(2k+1) and 2^(2k+2) - 1.

2, 8, 2048, 8796093022208, 2993155353253689176481146537402947624255349848014848

Offset: 1

Author

Michel Lagneau, Mar 28 2010

Keywords

• nonn

Comments

This tree represent the set of the interval [0,1], and the number of nodes of rank n is a(n). The number nodes with index 2^2k is a(2^2k) = (2^(2k+1) + 1)/3= 2^A007583.

Proof :

Let n(k) such that a(2^2k) = 2^n(k). By recurrence we have n(k) = 2^2k - 2^2k-1 + n(k-1) = 2^2k-1 + n(k-1). With n(1) = 3 = 2+1 we obtain : n(k) = 1 + 2(1 + 2^2 + ... + 2^(2k-2)) = (2^(2k+1) + 1)/3.

Graphic representation :

0....................^

1............l.................l

2............^.................^

3.......^........^........^........^

4.....l...l... l...l... l...l... l...l

5.....l...l... l...l... l...l... l...l

6.....l...l... l...l... l...l... l...l

7.....l...l... l...l... l...l... l...l

8.....^...^... ^...^... ^...^... ^...^

9....^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

10..^^^^^^^^ ^^^^^^^^ ^^^^^^^^ ^^^^^^^^

Examples

			k=0, a(1) = 2 ; k=1, a(4) = 8 ; k=2, a(16) = 2048 ; k=3, a(64) = 2^43 = 8796093022208.
		

References

• J. E. Hutchinson, Fractal and self-similarity, Ind. U. Math. J. 30 (1981), 713-747.

Programs

Maple

for n from 0 to 5 do: p:=(2*4^n + 1)/3:q:=2^p:print(q):od:

Formula

a(2^2k) = 2^A007583.