|
|
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
|
|
|
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():
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|