A309167 - OEIS

%I #29 Dec 09 2024 17:28:20

%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 345725,457813,576757,755173,806885,839285,924157

%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_1.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

%E a(29)-a(32) from _Rémy Sigrist_, Nov 16 2020