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A125051 The sub-Fibonacci tree; a rooted tree in which every node with label k and parent node with label g has g child nodes that are assigned labels beginning with k+1 through k+g; the tree starts at generation n=0 with a root node labeled '1' and a child node labeled '2'. 3
1, 2, 3, 4, 5, 5, 6, 7, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 11, 7, 8, 9, 10, 11, 8, 9, 10, 11, 12, 9, 10, 11, 12, 13, 7, 8, 9, 10, 11, 8, 9, 10, 11, 12, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 8, 9, 10, 11, 12, 13, 9, 10, 11, 12, 13, 14, 10, 11, 12, 13, 14, 15, 11, 12, 13, 14, 15, 16, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The maximum label for nodes in generation n is Fibonacci(n+2) for n>=0. The total number of nodes in generation n equals A005270(n+2) for n>=0. The sum of the labels for nodes in generation n equals A125052(n).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..24903 (generations 0..8)

Peter C. Fishburn and Fred S. Roberts, Elementary sequences, sub-Fibonacci sequences, Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.

EXAMPLE

The initial nodes of the tree for generations 0..5 are:

gen.0: [1];

gen.1: [2];

gen.2: [3];

gen.3: [4,5];

gen.4: (4)->[5,6,7],(5)->[6,7,8];

gen.5: (5)->[6,7,8,9],(6)->[7,8,9,10],(7)->[8,9,10,11],

(6)->[7,8,9,10,11],(7)->[8,9,10,11,12],(8)->[9,10,11,12,13].

By definition, there are 2 child nodes for node [3] of gen.2 since the parent of node [3] has label 2;

likewise, there are 3 child nodes for nodes [4] and [5] of gen.3 since the parent of both nodes has label 3.

The number of nodes in generation n begins:

1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368, 13067353, ...

The sum of the labels for nodes in generation n begins:

1, 2, 3, 9, 39, 252, 2361, 32077, 631058, 18035534, ...

MAPLE

g:= proc(n) option remember; `if`(n=0, [[1, 1]],

map(x-> seq([x[2], x[2]+i], i=1..x[1]), g(n-1)))

end:

T:= n-> map(x-> x[2], g(n)):

a:= proc() local i, l; i, l:= -1, []; proc(n) while

nops(l)<=n do i:=i+1; l:=[l[], T(i)[]] od; l[n+1] end

end():

seq(a(n), n=0..200); # Alois P. Heinz, Feb 08 2013

CROSSREFS

Cf. A005270, A125052.

Sequence in context: A097873 A005375 A138370 * A064067 A202306 A275579

Adjacent sequences: A125048 A125049 A125050 * A125052 A125053 A125054

KEYWORD

nonn,look

AUTHOR

Paul D. Hanna, Nov 19 2006

STATUS

approved

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Last modified December 7 05:41 EST 2022. Contains 358649 sequences. (Running on oeis4.)