Sub-Fibonacci Tree

[Paul D. Hanna]

Seqfans,
    Here I extend A005270 by one more term and also extend the sequence 
a(n) = sum of labels of nodes in generation n in the sub-Fibonacci tree.
I demonstrate a slightly faster method to compute these sequences. 
 
First, I am changing the description to be (hopefully) more clear: 
 
"Sum of labels of nodes in generation n of the sub-Fibonacci tree in
which 
a node with label k and parent node with label g, has g child nodes that 
are given 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'."
 
Would appreciate any suggestions to change in wording.
  
 
EXTENDING A005270.
There is a faster way to compute b(n)=A005270(n+1): 
b(n) = "number of nodes in generation n of the sub-Fibonacci tree"
 
by employing the following statistic:
 
b(n+1) = sum of (parent label)*(label) over all nodes in generation n. 
 
For example, 
b(4) = 27 = 3*(4+5);
b(5) = 177 = 4*(5+6+7) + 5*(6+7+8);
b(6) = 1680 = 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); ...

Using this method, I extend sequence A005270 by one more term:
1, 1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368, 13067353,
 

SUM OF LABELS SEQUENCE. 
In like manner, the computation of:
a(n) = "sum of labels of nodes in generation n of the sub-Fibonacci tree"

can be accomplished by the statistic:
 
a(n+1) = sum of (parent label)*(label) over all nodes in generation n 
+ sum of (parent label)*[label*(label+1)/2] over all nodes in generation
n-1. 
 
For example, 
a(2) = 3 = 1*2  + 1*( 1*2/2 ) ;
a(3) = 9 = 2*3  + 1*( 2*3/2 ) ;
a(4) = 39 = 3*(4+5)  + 2*( 3*4/2 ) ;
a(5) = 252 = 4*(5+6+7) + 5*(6+7+8)  + 3*( 4*5/2 + 5*6/2 ) ;
a(6) = 2361 = 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)  +
 4*( 5*6/2 + 6*7/2 + 7*8/2 ) + 5*( 6*7/2 + 7*8/2 + 8*9/2) ;
...
 
With this method, I extend the sum of labels in generation n to: 
1, 2, 3, 9, 39, 252, 2361, 32077, 631058, 18035534, 
 

Would someone please confirm my calculation of the terms: 
b(10) = A005270(11) = 13067353  and 
a(10) = 18035534
 
and maybe compute a few more terms? 
Thanks,
    Paul
----------------------------------------------------
> 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] since 
> the parent of node [3] is [2]; likewise, 
> there are 3 child nodes for nodes [4] and [5] (in gen.3) 
> since the parent of both nodes has label [3] (in gen.2). 
>  
> Notice the minimum label in generation n is n+1, and 
> the maximum label in generation n is Fibonacci(n+2).  
>   
> From this tree I am interested in gleaning 2 sequences. 
>   
> SEQUENCE 1. 
> I compute the number of nodes in generation n to begin: 
> [1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368, ...]
>  
> which is equal (with offset) to A005270:
> "Sub-Fibonacci sequences of length n."
> 1, 1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368 
>  
> SEQUENCE 2. 
> But the following related sequence is NOT found in the OEIS: 
>  
> Sum of labels of nodes in generation n:
> [1, 2, 3, 9, 39, 252, 2361, 32077, 631058, ...]
>  
> EXTENSION?
> Could someone compute more terms of sequences 1 and 2? 
>  
> Thanks,
>     Paul