A348850 a(n) is the number of labeled rooted unordered binary trees T where the nodes are labeled with distinct positive integers, the root has label n, each parent label equals the sum of its children labels, and T cannot be extended.

1, 1, 1, 1, 2, 2, 3, 5, 9, 10, 14, 22, 33, 57, 66, 94, 132, 188, 317, 454, 576, 806, 1083, 1535, 2342, 3215, 5231, 5656, 8545, 10804, 15226, 21153, 30342, 44536, 63165, 73877, 107241, 133994, 178497, 247564, 331695, 472331

Offset: 1

Author

Rémy Sigrist, Nov 01 2021

Keywords

• nonn
• more

Comments

For any n > 0:

- we can imagine a variant of Grundy's game where we start with n at root position,

- and each move consists in adding to a leaf, say w, two children, u and v such that 0 < u < v and u+v = w and u and v do not already appear in the tree,

- a(n) gives the number of final positions (where no move is possible).

Examples

			For n = 1, 2, 3, 4: a(n) = 1:
          |         |         |         |
          1         2         3         4
                             / \       / \
                            1   2     1   3
For n = 5, 6: a(n) = 2:
          |         |         |         |
          5         5         6         6
         / \       / \       / \       / \
        1   4     2   3     1   5     2   4
                               / \       / \
                              2   3     1   3
		

Links

• Rémy Sigrist, PARI program for A348850
• Wikipedia, Grundy's game

Crossrefs

Cf. A124973, A346523, A348851.

Programs

PARI

See Links section.