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%I #6 Apr 03 2014 11:35:08
%S 1,1,3,18,180,4347,245511,33731424,11850958449,10823718435525,
%T 26127739209077469,169071160476526474689,2962647736390311022542681,
%U 141814999458311839862777779311,18682218330844513414826192858258922
%N Number of sequences {c(i), i=0..n} that form the initial terms of a self-convolution cube of an integer sequence such that 0 < c(n) <= 3*c(n-1) for n>0 with c(0)=1.
%C Equals the number of nodes at generation n in the 3-convoluted tree. The minimal path in the 3-convoluted tree is A083953 and the maximal path is A132835. The 3-convoluted tree is defined as follows: tree of all finite sequences {c(k), k=0..n} that form the initial terms of a self-convolution cube of some integer sequence such that 0 < c(n) <= 3*c(n-1) for n>0 with a(0)=1.
%H Martin Fuller, <a href="/A132852/a132852.txt">Computing A132852, A132853, A132854, A132855, A132856</a>
%e a(n) counts the nodes in generation n of the following tree.
%e Generations 0..4 of the 3-convoluted tree are as follows;
%e The path from the root is shown, with child nodes enclosed in [].
%e GEN.0: [1];
%e GEN.1: 1->[3];
%e GEN.2: 1-3->[3,6,9];
%e GEN.3:
%e 1-3-3->[1,4,7]
%e 1-3-6->[1,4,7,10,13,16]
%e 1-3-9->[1,4,7,10,13,16,19,22,25];
%e GEN.4:
%e 1-3-3-1->[3]
%e 1-3-3-4->[3,6,9,12]
%e 1-3-3-7->[3,6,9,12,15,18,21]
%e 1-3-6-1->[3]
%e 1-3-6-4->[3,6,9,12]
%e 1-3-6-7->[3,6,9,12,15,18,21]
%e 1-3-6-10->[3,6,9,12,15,18,21,24,27,30]
%e 1-3-6-13->[3,6,9,12,15,18,21,24,27,30,33,36,39]
%e 1-3-6-16->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48]
%e 1-3-9-1->[3]
%e 1-3-9-4->[3,6,9,12]
%e 1-3-9-7->[3,6,9,12,15,18,21]
%e 1-3-9-10->[3,6,9,12,15,18,21,24,27,30]
%e 1-3-9-13->[3,6,9,12,15,18,21,24,27,30,33,36,39]
%e 1-3-9-16->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48]
%e 1-3-9-19->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57]
%e 1-3-9-22->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66 ]
%e 1-3-9-25->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75].
%e Each path in the tree from the root node forms the initial terms of
%e a self-convolution cube of a sequence of integer terms.
%Y Cf. A132852, A132854, A132855, A132856.
%Y Cf. A083953, A132835.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 19 2007, Oct 06 2007
%E Extended by _Martin Fuller_, Sep 24 2007.
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