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%I #28 Mar 12 2020 00:25:54
%S 0,0,3,24,1680,5317635,66314914699608,8947678119828215014722891024,
%T 178098260698995011212395018312912894502905113202338936835
%N The number of independent sets in a complete binary tree with n levels (i.e., with 2^n-1 vertices) in which every vertex has degree 3.
%C For example, when n=3, there are two degree 3 vertices which do not share an edge. There are then three degree 3 (regular) independent subsets so a(3)=3. a(10) has 113 digits and is too large to include. The sequence is related to A076725, the number of independent sets in a complete binary tree.
%C The independent sets sought are those in the subgraph induced by the degree-3 vertices. This subgraph is a forest comprising two complete binary trees with n-2 levels each. These trees have A076725(n-2+1) independent sets each and the empty set (empty in both) is excluded here so a(n) = A076725(n-1)^2 - 1. - _Kevin Ryde_, Mar 10 2020
%F a(n) = (a(n-2) + 1)^4 + 2(a(n-1)+1)(a(n-2) + 1)^2 + (a(n-1) + 1)^2 - 1, with a(1) = a(2) = 0.
%e a(3) = (a(1) + 1)^4 + 2(a(2)+1)(a(1) + 1)^2 + (a(2) + 1)^2 - 1 = (0+1)^4+2(0+1)(0+1)^2+(0+1)^2-1 = 3.
%e a(4) = (a(2) + 1)^4 + 2(a(3)+1)(a(2) + 1)^2 + (a(3) + 1)^2 - 1 = (0+1)^4+2(3+1)(0+1)^2+(3+1)^2-1 = 24.
%e a(5) = (a(3) + 1)^4 + 2(a(4)+1)(a(3) + 1)^2 + (a(4) + 1)^2 - 1 = (3+1)^4+2(24+1)(3+1)^2+(24+1)^2-1 = 1680.
%o (PARI) a(n) = my(x=1,y=1); for(i=3,n, [x,y] = [(x + y^2)^2, x]); x-1; \\ _Kevin Ryde_, Mar 10 2020
%Y Cf. A076725.
%K nonn
%O 1,3
%A _Steven Kuipers_ and _Bradford M. Morris_, May 04 2014
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