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The tripent is equal to $$\lbrace5,5,5\rbrace$$ or $$5\lbrace5\rbrace5$$ in BEAF, $$5\uparrow\uparrow\uparrow\uparrow\uparrow5$$ in arrow notation, or 5 heptated to 5.[1]. The term was coined by Jonathan Bowers. This is the fifth Ackermann number.

Googology Wiki user Hyp cos calls this number a trienpent, and it's equal to s(5,5,5), s(5,2,6), s(5,5,1,2), s(5,1,1,3), or s(5,1,1,1,2) in strong array notation. [2]

It can be computed during the following process:

1. A(1) = 5
2. A(2) = 5^5^5^5^5 (A(1) 5's)
3. A(3) = 5^5^5^...5^5^5, with A(2) 5's.
4. A(4) = 5^5^5^5^5^......5^5^5^5, with A(3) 5's
5. Continue, let A(n) be 5^5^5^5^......^5^5^5 with A(n-1) 5's
6. Tripent is A(A(A(A(A(A(.........A(A(A(A(1)))).........)))))), where there are A(A(A(A(A(......A(A(A(1)))......))))) A's, where there are A(A(A(A(......A( A(A(1)))......)))) A's, where there are A(A(A(A(A(1))))) A's.

## Etymology

The name of this number is based on tri- (three) and penta- (five), meaning that there are 3 5's in the array.

## Approximations

Notation Approximation
Up-arrow notation $$5\uparrow\uparrow\uparrow\uparrow\uparrow5$$ (exact)
BEAF $$\{5,5,5\}$$ (exact)
Hyper-E notation $$E(5)1\#1\#1\#1\#5$$ (exact)
Chained arrow notation $$5\rightarrow5\rightarrow5$$ (exact)
Hyperfactorial array notation $$6!4$$
Fast-growing hierarchy $$f_6(4)$$
Hardy hierarchy $$H_{\omega^6}(4)$$
Slow-growing hierarchy $$g_{\varphi(4,0)}(5)$$

## Sources

1. Bowers, JonathanInfinity Scrapers. Retrieved January 2013.
2. Numbers from linear array notation | Steps Toward Infinity!

Tritri series: tritri · tritet · tripent · trisept · tridecal · grand tridecal
Tetratri series: tetratri · supertet · general(plex)
Pentatri series: pentatri · superpent · pentadecal(plex)
Hexatri series: hexatri · superhex · hexadecal(plex)
Heptatri series: heptatri · supersept · heptadecal
Iteral series: superoct · octadecal · superenn · ennadecal · iteral · ultatri
Dupertri series: dupertri · duperdecal · truperdecal · quadruperdecal
Latri series: latri · emperal(plex) · hyperal(plex) · admiral
Dutritri series: dutritri · dutridecal
Dimentri series: dimentri · dulatri · trilatri · trimentri
Triakulus series: triakulus · tridecatrix
Big boowa series: big boowa · great big boowa · grand boowa
Tiaokhiao's extensions: trihex · trioct · triennet · triundecal · tridodecal · tritriplex · tritriplexian · grand tritri · tetrapent · tetrahex · grand tetratri · octatri · enneatri · decatri · undecatri · dodecatri