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A309167 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. 2

%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^(n-1).

%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

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Last modified January 21 13:55 EST 2020. Contains 331113 sequences. (Running on oeis4.)