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A082007 Triangle (an infinite binary tree) read by rows; see Comments lines for definition. 4
0, 1, 2, 3, 6, 9, 12, 4, 5, 7, 8, 10, 11, 13, 14, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 16, 17, 31, 32, 46, 47, 61, 62, 76, 77, 91, 92, 106, 107, 121, 122, 136, 137, 151, 152, 166, 167, 181, 182, 196, 197 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

...

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)

Moreover, the first 2^(2^n) - 1 terms are a permutation of the first 2^(2^n) - 1 nonnegative integers. - Ivan Neretin, Mar 12 2017

REFERENCES

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

LINKS

Ivan Neretin, Table of n, a(n) for n = 0..8190

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

EXAMPLE

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.

MATHEMATICA

w = {{0}}; Do[k = 2^Floor@Log2[n - 1]; AppendTo[w, Flatten@Table[w[[n - k]] + (2^k - 1) i, {i, 2^k}]], {n, 2, 7}]; a = Flatten@w (* Ivan Neretin, Mar 12 2017 *)

PROG

(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 )

# Steve Witham (sw(AT)tiac.net), Oct 11 2009

CROSSREFS

Cf. A082008, A082009.

Sequence in context: A018331 A032704 A029805 * A064417 A191981 A131975

Adjacent sequences:  A082004 A082005 A082006 * A082008 A082009 A082010

KEYWORD

nonn,tabf,easy

AUTHOR

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

STATUS

approved

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Last modified April 20 03:36 EDT 2019. Contains 322294 sequences. (Running on oeis4.)