0,3

At stage 0 we begin with the triangle L_0

.........................0

........................1.2

This has 2 nodes on the lowest level, and 2^2-1 nodes in total.

At stage 1 we construct L_1 by adding 2^2 copies of L_0 to

the lowest level nodes in L_0. Thus L_1 has 3+12 = 2^4-1

nodes in total (labeled 0 to 14), with 2^3 nodes at the lowest level.

At stage 2 we construct L_2 by adding 2^4 copies of L_1 to

the lowest level nodes in L_1. Thus L_2 has 15+240 = 2^8-1

nodes in total (labeled 0 to 254), with 2^(2^3-1) nodes at the lowest level.

...

At stage k we construct L_k by adding 2^(2^k) copies of L_(k-1) to

the lowest level nodes in L_(k-1). Thus L_k has 2^(2^(k+1))-1

nodes in total (labeled 0 to 2^(2^(k+1))-2), with 2^(2^(k+1)-1) nodes at the lowest level.

...

Comment from Steve Witham, Oct 08 2009: This is a special case of what's called the "Van Emde Boas layout" - see p. 203 0f the Meyer et al. reference. "Split the tree in the middle, at height h/2. This breaks the tree into a top recursive subtree of height floor(h/2) and several bottom subtrees of height ceil(h/2). There are sqrt(N) bottom subtrees, each of size sqrt(N)."

Contribution from Steve Witham (sw(AT)tiac.net), Oct 13 2009: (Start)

Starting the sequence (and its index) at 1 (as in A082008) instead of 0 (as in A082007) seems more natural. This was conceived as a way to arrange a heapsort in memory to improve locality of reference. The classic Williams/Floyd heapsort also works a little more naturally when the origin is 1.

This sequence is a permutation of the integers >= 0. (End)

Ulrich Meyer, Peter Sanders and Jop Sibeyn, Algorithms for Memory Hierarchies: Advanced Lectures.

Steve Witham, Clumpy Heapsort

Steve Witham, Clumpy Heapsort. [From Steve Witham (sw(AT)tiac.net), Oct 13 2009]

The beginning of the tree:

.....................................0

....................................1.2

............................3.....6......9.....12

..........................4...5..7.8...10.11.13..14

............15.......................30......................45.......240

..........16..17...................31..32..................46..47...241.242

...18....21...24....27......33....36...39....42......48....51...54....57

19.20.22.23.25.26.28.29..34.35.37.38.40.41.43.44..49.50.52.53.55.56.58.59

etc.

(15 and 30 are children of 4, 45 and 60 are children of 5, and so on)

Rows 0 and 1 form L_0, rows 0 through 3 form L_1, rows 0 through 7 form L_2, and so on.

Contribution from Steve Witham (sw(AT)tiac.net), Oct 11 2009: (Start)

(Python)

def A082007( n ):

..if n == 0: return 0

..

..y = 2 ** int( log( n + 1, 2 ) )

..yc = 2 ** 2 ** int( log( log( y, 2 ), 2 ) )

..yr = y / yc

..return (yc-1) * int( (n+1-y)/yr + 1 ) + A082007( yr + (n+1)%yr - 1 )

(End)

Cf. A082008, A082009.

Sequence in context: A018331 A032704 A029805 this_sequence A064417 A131975 A057474

Adjacent sequences: A082004 A082005 A082006 this_sequence A082008 A082009 A082010

nonn,tabf,easy

N. J. A. Sloane (njas(AT)research.att.com), Oct 06 2009, based on a posting by Steve Witham (sw(AT)tiac.net) to the Math Fun Mailing List, Sep 30 2009

Added Python function. [From Steve Witham (sw(AT)tiac.net), Oct 11 2009]