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

%I A300402 #17 Feb 16 2025 08:33:53
%S A300402 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,
%T A300402 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,
%U A300402 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
%N A300402 Smallest integer i such that TREE(i) >= n.
%C A300402 The sequence grows very slowly.
%C A300402 A rooted tree is a tree containing one special node labeled the "root".
%C A300402 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
%H A300402 Priyabrata Biswas, <a href="https://towardsdatascience.com/how-big-is-the-number-tree-3-61b901a29a2c">Towards Data Science: How Big Is The Number — Tree(3)</a>
%H A300402 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RootedTree.html">Rooted Tree</a>
%H A300402 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperoperation#Notations">Hyperoperation - Notations</a>
%H A300402 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem">Kruskal's tree theorem</a>
%e A300402 TREE(1) = 1, so a(n) = 1 for n <= 1.
%e A300402 TREE(2) = 3, so a(n) = 2 for 2 <= n <= 3.
%e A300402 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).
%Y A300402 Cf. A090529, A300403, A300404.
%Y A300402 Labeled rooted trees are counted by A000169 and A206429.
%Y A300402 Cf. A000081, A000311, A060313, A060356, A317713.
%K A300402 nonn
%O A300402 0,3
%A A300402 _Felix Fröhlich_, Mar 05 2018