Explosion

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Explosion refers to the binary function \(a\ \{\{\{1\}\}\}\ b = \{a,b,1,3\}\), using BEAF.^[1]

Intuitively, explosion can be defined like so:

\[a\ \{\{\{1\}\}\}\ b = a\ \{\{a\ \{\{a \ldots \{\{a\}\} \ldots a\}\}\ a\}\}\ a\]

with \(b\) copies of \(a\) from the center out.

In the fast-growing hierarchy, \(f_{\omega2 +1}(n)\) approximately corresponds to explodal growth rate.

Examples[]

\(\{10,2,1,3\} = \{10,10,10,2\}\) (grand tridecal)
\(\{10,100,1,3\} = \{10,10,\{10,99,1,3\},2\}\) (Dukil-Googol, Cookiefonster called the number corplodal)

Pseudocode[]

Below is an example of pseudocode for explosion.

function explosion(a, b):
    result := a
    repeat b - 1 times:
         result := hyperexpansion(a,a,result)
    return result

function hyperexpansion(a, b, n):
    result := a
    repeat b - 1 times:
        if n = 1:
            result := hyper(a,a,result+2)
        else:
            result := hyperexpansion(a, result, n - 1)
    return result

function hyper(a, b, n):
    if n = 1:
        return a + b
    result := a
    repeat b - 1 times:
        result := hyper(a, result, n - 1)
    return result