A300402 Smallest integer i such that TREE(i) >= n.

1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Author

Felix Fröhlich, Mar 05 2018

Keywords

• nonn

Comments

The sequence grows very slowly.

A rooted tree is a tree containing one special node labeled the "root".

TREE(n) gives the largest integer k such that a sequence T(1), T(2), ..., T(k) of vertex-colored (using up to n colors) rooted trees, each one T(i) having at most i vertices, exists such that T(i) <= T(j) does not hold for any i < j <= k. - Edited by Gus Wiseman, Jul 06 2020

Examples

			TREE(1) = 1, so a(n) = 1 for n <= 1.
TREE(2) = 3, so a(n) = 2 for 2 <= n <= 3.
TREE(3) > A(A(...A(1)...)), where A(x) = 2[x+1]x is a variant of Ackermann's function, a[n]b denotes a hyperoperation and the number of nested A() functions is 187196, so a(n) = 3 for at least 4 <= n <= A^A(187196)(1).
		

Links

• Priyabrata Biswas, Towards Data Science: How Big Is The Number — Tree(3)
• Eric Weisstein's World of Mathematics, Rooted Tree
• Wikipedia, Hyperoperation - Notations
• Wikipedia, Kruskal's tree theorem

Crossrefs

Cf. A090529, A300403, A300404.

Labeled rooted trees are counted by A000169 and A206429.

Cf. A000081, A000311, A060313, A060356, A317713.