H function

H(n) is an extremely fast-growing function devised by Chris Bird.

Using Bird's Nested Hyper-Nested array notation, \(H(n) = \{3,n [1 [1 [1 [ \cdots [1 [1 \_n 1 \_n 2] 2] \cdots ] 2] 2] 2] 2\}\) (with n sets of square brackets).

First few values are shown below:

\(H(1) = \{3,1 [1 \ 1 \ 2] 2\} = 3\)

\(H(2) = \{3,2 [1 [1 \neg 1 \neg 2] 2] 2\}\)

\(H(3) = \{3,3 [1 [1 [1 \_3 1 \_3 2] 2] 2] 2\}\)

\(H(4) = \{3,4 [1 [1 [1 [1 \_4 1 \_ 4 2] 2] 2] 2] 2\}\)

In his "Bowers Named Numbers" article, Bird stated that his \([1 [1 \neg 1 \neg 2] 2]\) separator has relationship between Bowers' forward slash one as follows:

\(\{b,p+1 / 2\} < \{b,p [1 [1 \neg 1 \neg 2] 2] 2\} < \{b,p+2 / 2\}\)

Thus, H(2) can be bounded in BEAF:

\(\{3,3 / 2\} < H(2) < \{3,4 / 2\}\).