cp's OEIS Frontend

A125052 Sum of labels for nodes in generation n of the sub-Fibonacci tree (A125051).

%I A125052 #12 Mar 03 2020 09:58:22
%S A125052 1,2,3,9,39,252,2361,32077,631058,18035534,751936149,45973362492,
%T A125052 4144777181393,554100538432001,110435083963283354,32981178674724868365
%N A125052 Sum of labels for nodes in generation n of the sub-Fibonacci tree (A125051).
%C A125052 The sub-Fibonacci tree is 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'. The number of nodes in generation n of the sub-Fibonacci tree is A005270(n+2); the maximum label in generation n is Fibonacci(n+2).
%H A125052 Peter C. Fishburn and Fred S. Roberts, <a href="https://doi.org/10.1016/0166-218X(93)90236-H">Elementary sequences, sub-Fibonacci sequences</a>, Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.
%e A125052 The initial nodes of the sub-Fibonacci tree for generations 0..5 are:
%e A125052 gen.0: [1];
%e A125052 gen.1: [2];
%e A125052 gen.2: [3];
%e A125052 gen.3: [4,5];
%e A125052 gen.4: (4)->[5,6,7],(5)->[6,7,8];
%e A125052 gen.5: (5)->[6,7,8,9],(6)->[7,8,9,10],(7)->[8,9,10,11],
%e A125052 (6)->[7,8,9,10,11],(7)->[8,9,10,11,12],(8)->[9,10,11,12,13].
%e A125052 The sum of the labels for nodes in generation n+1 >= 2 is equal to:
%e A125052 a(n+1) = sum (parent label)*(label) over all nodes in generation n + sum (parent label)*[label*(label+1)/2] over all nodes in gen. n-1.
%e A125052 For example:
%e A125052 a(2) = 3 = 1*2 + 1*( 1*2/2 );
%e A125052 a(3) = 9 = 2*3 + 1*( 2*3/2 );
%e A125052 a(4) = 39 = 3*(4+5) + 2*( 3*4/2 );
%e A125052 a(5) = 252 = 4*(5+6+7) + 5*(6+7+8) + 3*( 4*5/2 + 5*6/2 );
%e A125052 a(6) = 2361 = 5*(6+7+8+9) + 6*(7+8+9+10) + 7*(8+9+10+11) +
%e A125052 6*(7+8+9+10+11) + 7*(8+9+10+11+12) + 8*(9+10+11+12+13) +
%e A125052 4*( 5*6/2 + 6*7/2 + 7*8/2 ) + 5*( 6*7/2 + 7*8/2 + 8*9/2 ).
%Y A125052 Cf. A125051, A005270.
%K A125052 nonn
%O A125052 0,2
%A A125052 _Paul D. Hanna_, Nov 19 2006
%E A125052 a(10)-a(15) from _Alois P. Heinz_, May 03 2015