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

1,2

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.

Links

Table of n, a(n) for n=1..32.

Rémy Sigrist, Illustration of first terms

Rémy Sigrist, C++ program for A309167

Formula

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

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

Example

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
                  |

Prog

(C++) See Links section.

Crossrefs

Cf. A000351, A309228.

Sequence in context: A149573 A149574 A301634 * A272069 A018678 A149575

Adjacent sequences: A309164 A309165 A309166 * A309168 A309169 A309170

Keyword

nonn,more

Author

Rémy Sigrist, Jul 15 2019

Extensions

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

Status

approved