%I
%S 1,5,13,65,97,229,997,1145,2245,5725,7213,9805,10445,24193,34121,
%T 37321,52225,83729,98449,125233,145493,156925,171037,260893,334981,
%U 345725,457813,576757
%N a(n)^2 is the least possible value at the root of a binary tree of height n where all nodes hold positive squares and all interior nodes also equal the sum of their two children.
%C We have binary trees with the desired properties for every height n > 0:
%C  for n = 1: we have the following tree B_1:
%C 1^2
%C 
%C  for any n > 0, provided we have B_n, we can build a tree B_{n+1) as follows:
%C 3^2*B_n 4^2*B_n
%C \ /
%C \ /
%C \ /
%C (5^n)^2
%C 
%C  hence the sequence is well defined.
%H Rémy Sigrist, <a href="/A309167/a309167.png">Illustration of first terms</a>
%H Rémy Sigrist, <a href="/A309167/a309167.txt">C++ program for A309167</a>
%F a(n) <= 5^(n1).
%F A309228(a(n)) = n and A309228(k) < n for any k < a(n).
%e a(1) = 1:
%e 1^2
%e 
%e a(2) = 5:
%e 3^2 4^2
%e \ /
%e \ /
%e 5^2
%e 
%e a(3) = 13:
%e 3^2 4^2
%e \ /
%e \ /
%e 5^2 12^2
%e \ /
%e \ /
%e 13^2
%e 
%o (C++) See Links section.
%Y Cf. A000351, A309228.
%K nonn,more
%O 1,2
%A _Rémy Sigrist_, Jul 15 2019
