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.

1, 5, 13, 65, 97, 229, 997, 1145, 2245, 5725, 7213, 9805, 10445, 24193, 34121, 37321, 52225, 83729, 98449, 125233, 145493, 156925, 171037, 260893, 334981, 345725, 457813, 576757, 755173, 806885, 839285, 924157

Offset: 1

Author

Rémy Sigrist, Jul 15 2019

Keywords

• nonn
• more

Comments

We have binary trees with the desired properties for every height n > 0:

- for n = 1: we have the following tree B_1:

1^2

|

- for any n > 0, provided we have B_n, we can build a tree B_{n+1} as follows:

3^2*B_n 4^2*B_n

\ /

\ /

\ /

(5^n)^2

|

- hence the sequence is well defined.

Examples

			a(1) = 1:
              1^2
               |
a(2) = 5:
           3^2    4^2
            \     /
             \   /
              5^2
               |
a(3) = 13:
          3^2    4^2
           \     /
            \   /
             5^2    12^2
              \      /
               \    /
                13^2
                  |
		

Links

• Rémy Sigrist, Illustration of first terms
• Rémy Sigrist, C++ program for A309167

Crossrefs

Cf. A000351, A309228.

Formula

a(n) <= 5^(n-1).

A309228(a(n)) = n and A309228(k) < n for any k < a(n).

Extensions

a(29)-a(32) from Rémy Sigrist, Nov 16 2020