## FANDOM

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The Supernova Array is a method of creating very large numbers using recursion of Up-Arrow Notation in a simple linear notation. It is loosely based on the Ultra-Factorial Funcional Array.

## Definitions and Examples

• Base Function: $$S(a,b) = a\uparrow^{(b)}a$$, where $$b$$ is the number of $$\uparrow$$'s.
• $$S(a,b,0) = S(a,b)$$
• $$S(a,b,c) = S(a,S(a,b),c-1)$$
• $$S(a,b,c,0) = S(a,b,c)$$
• $$S(a,b,c,d) = S(a,b,S(a,b,c),d-1)$$
• $$S(a,b,c,...x,y,z) = S(a,b,c,...x,S(a,b,c,...x,y),z-1)$$

In general, replace the second to the last input ($$y$$) with the entire function excluding the last input ($$z$$), then subtract 1 from the last input. Do the same for the function within, and the subsequent function, and the function after that, etc. until you reach the base function $$S(a,b)$$.

$$S(1,1) = 1\uparrow 1 = 1$$

$$S(2,2) = 2\uparrow \uparrow 2 = 4$$

$$S(3,3) = 3\uparrow \uparrow \uparrow 3$$ = tritri

$$S(10,10) = 10\uparrow^{(10)} 10$$ = tridecal

$$S(3,1,1) = S(3,S(3,1),0) = S(3,S(3,1)) = S(3,3\uparrow 3) = S(3,27)$$

$$S(3,3,1) = S(3,S(3,3)) = S(3,tritri)$$

$$S(10,10,1) = S(10,S(10,10)) = S(10,tridecal)$$

$$S(3,3,3) = S(3,S(3,3),2) = S(3,tritri,2) = S(3,S(3,tritri),1) = S(3,S(3,S(3,tritri)),0)$$

$$S(3,4,63) = S(3,S(3,4),62) = S(3,S(3,S(3,4)),61)...$$ = Graham's Number

## Extended Notations

In extended notations, parentheses are added into the array in order to apply systems of recursion.

• $$S((a,b)) = S(a,a,a,...a)$$, where there are $$b$$ number of $$a$$'s
• $$S((a,b,c,...x,y,z)) = S((a,b,c,...x,S(a,b,c,...x,y),z-1))$$
• $$S(((a,b))) = S((a,a,a...,a))$$, where there are $$b$$ number of $$a$$'s
• $$S(((a,b,c,...x,y,z))) = S(((a,b,c,...x,S(a,b,c,...x,y),z-1)))$$

In general, $$S(a,b)_n = S(a,a,a,...a)_{n-1}$$, where there are $$b$$ number of $$a$$'s and $$n$$ number of paretheses. Alternatively, $$S(((a,b)))$$ can be written as $$S(a,b)_3$$.

There may be cases wherein a simplified notation may be desirable, as in $$S(a,b)_{S(a,b)}$$. In this case, a subscript is added to S, such that the aforementioned function can be rewritten as $$S_1(a,b)$$. In general,

• $$S_0(a,b) = S(a,b)$$
• $$S_{(n+1)}(a,b) = S(a,b)_{(S_n(a,b))}$$, where there are $$S_n(a,b)$$ number of parentheses in $$S(((...(a,b)))...)$$
• $$S_n(a,b,c,...x,y,z) = S_n(a,b,c,...x,S(a,b,c,...x,y),z-1)$$

Likewise, a simplified notation may be desired for functions such as $$S_{S(a,b)}(a,b)$$. A superscript is instead assigned to S, such that the aforementioned function may be rewritten as $$S^1(a,b)$$. In general,

• $$S^0(a,b) = S(a,b)$$
• $$S^{(n+1)}(a,b) = S_{(S^n(a,b))}(a,b)$$
• $$S^n(a,b,c,...x,y,z) = S^n(a,b,c,...x,S(a,b,c,...x,y),z-1)$$

Furthermore, $$S^{S(a,b)}(a,b)$$ could also use a more simplified notation. A subscript is then added BEFORE S, such that the aforementioned function may be rewritten as $$_1S(a,b)$$. In general,

• $$_0S(a,b) = S(a,b)$$
• $$_{(n+1)}S(a,b) = S^{(_nS(a,b))}(a,b)$$
• $$_nS(a,b,c,...x,y,z) = _nS(a,b,c,...x,S(a,b,c,...x,y),z-1)$$

Finally, a function such as $$_{(S(a,b))}S(a,b)$$ would also benefit from simplification. A superscript is then added BEFORE S, such that the aforementioned function may be rewritten as $$^1S(a,b)$$. In general,

• $$^0S(a,b) = S(a,b)$$
• $$^{(n+1)}S(a,b) = _{(^nS(a,b))}S(a,b)$$
• $$^nS(a,b,c,...x,y,z) = ^nS(a,b,c,...x,S(a,b,c,...x,y),z-1)$$

However, the function's limit has been reached with the pre-superscript. In order to further expand the function, a new more succint formulation has to be added - $$S(a,b)[n,k]$$.

• $$S(a,b)_n$$ can be rewritten as $$S(a,b)[n,0]$$,
• $$S_n(a,b)$$ can be rewritten as $$S(a,b)[n,1]$$,
• $$S^n(a,b)$$ can be rewritten as $$S(a,b)[n,2]$$, and so on.

In general,

• If $$k = 0$$, $$S(a,b)_n\ = S(a,b)[n,0] = S(a,a,a,...a)[n-1,0]$$, where there are $$b$$ number of $$a$$'s
• If $$k > 0$$ and $$n = 0$$, $$S(a,b)[0,k] = S(a,b)$$
• If $$k > 0$$ and $$n > 0$$, $$S(a,b)[n,k] = S(a,b)[S(a,b)[n-1,k],k-1]$$

Although this function can already produce extremely huge numbers, it can still be further extended as follows:

$$S(a,b)[n,k,0] = S(a,b)[n,k]$$

$$S(a,b)[n,k,p] = S(a,b)[n,S(a,b)[n,k],p-1]$$

$$S(a,b)[n,k,p,...x,y,z] = S(a,b)[n,k,p,...x,S(a,b)[n,k,p,...x,y],z-1]$$

$$S(a,b)[ [n,k] ] = S(a,b)[n,n,n,...n]$$, where there are $$k$$ number of $$n$$'s

$$S(a,b)[n,k][1,0] = S(a,b)[n,k][0,s] = S(a,b)[n,k]$$

$$S(a,b)[n,k][r,0] = S(a,b)[n,n,n,...n][r-1,0]$$, where there are $$k$$ number of $$n$$'s

$$S(a,b)[n,k][r,s] = S(a,b)[n,k][S(a,b)[n,k][r-1,s],s-1]$$

$$S(a,b)[n,k][r,s,0] = S(a,b)[n,k][r,s]$$

$$S(a,b)[n,k][r,s,t] = S(a,b)[n,k][r,S(a,b)[n,k][r,s],t-1]$$

$$S(a,b)[n,k][r,s,t,...x,y,z] = S(a,b)[n,k][r,s,t,...x,S(a,b)[n,k][r,s,t,...x,y],z-1]$$

$$S[0](a,b) = S(a,b)$$

$$S[\gamma](a,b) = S[\gamma -1](a,b)[a,a][a,a][a,a]...[a,a]$$, where there are $$b$$ number of $$a$$'s

$$S[\gamma](a,b)[n,k] = S[\gamma](a,b)[S(a,b)[n-1,k],k-1]$$

Etc.