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Triakulus is equal to \(3\&3\&3\) or \(\{ 3,3 /2 \}\) or \(\{L,1\}_{3,3}\) using the array of operator and BEAF.[1] Since the array of operator solves from left to right, \(3\&3\&3 = \{ 3,3,3 \} \& 3 \) = \(3 \uparrow\uparrow\uparrow 3 \& 3\), a pentational array. The structure of an array which defines triakulus is supposedly X^^^X and the array is intended to have tritri entries.

It must be remembered that although it has tritri entries, a triakulus is in no way the same thing as a linear array with tritri repetitions of 3. That would be dupertri, which is a much smaller number. This is because you have to group the terms in the arrays, as an array of size X+1 & 10, when the terms are grouped, is equal to {10, 10(1)10}. If the terms are ungrouped, the array will be a linear array of size 10+1, which will be {10, 10+1(1)2}, which is just a linear array with size 11. One of the major challenges behind defining this level of BEAF would be successfully conceive a representation of triakulus that captures all its tritri entries while setting it apart from a dupertri, as a dupertri is an ungrouped 3^^^3 & 3, while triakulus is X^^^X & 3, where the X's are grouped.

If it's well-defined, this would be the smallest number named by Jonathan Bowers which lies in the pentational-array range.[2] This is the first and the smallest Bowerism which is not well-defined. See Introduction to BEAF#Pentational arrays for more details of that level.

Approximations[]

Since it is not defined by now, the following approximation is just an expected value under climbing method (strong bound):

Notation Approximation
Bird's array notation \(\{3,3 [1 [3 ¬ 2] 1 [3 ¬ 2] 1 [3 ¬ 2] 2] 2\}\)
Extended Cascading-E notation E[3]3#^^#^^#3

= E[3]3#^^^#3

Fast-growing hierarchy \(f_{\varphi(\varepsilon_0,0)}(3)\)
Hardy hierarchy \(H_{\varphi(\varepsilon_0,0)}(3)\)

Sources[]

See also[]

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) · diteral(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
RedBandServerYT's extensions: tridecalplex
Douglas Shamlin Jr.'s extensions: quintuperdecal · triteral · dutetradecal · quadriteral · dupentadecal · duladecal · triladecal
MultiArrest91's extensions: tritredecal · tritetradecal · sextuperdecal · septuperdecal · octuperdecal · nonuperdecal · decuperdecal · centuperdecal · milluperdecal · quadritetradecal · quintitridecal
SeveralLegend9998's extensions: corboral
XRQ CORPORATION's extensions: quintiteral · quadrilatri · quintilatri · sextilatri · septilatri · octilatri · nonilatri · decilatri
Exeen Management World's extensions: dutritet · dutetratet · dupentapent · duheptasept · duheptadecal · duoctadecal · duennadecal · grand dutritri · tritridecal · tritetradecal · quadritridecal
BAT4423's extensions: duhexadecal

Jonathan Bowers' array of series

Goppatoth group: Goppatoth · Goppatothplex
Pentation level: Triakulus · Kungulus · Kungulusplex
Higher operation group: Quadrunculus · Tridecatrix · Humongulus
Lineational group: Tetdecatrix · Pendecatrix · Hexdecatrix · Hepdecatrix · Ocdecatrix · Endecatrix · Lineatrix · Lineatrixplex
Golapulus group: Golapulus · Golapulusplex · Dekulus
Cookie Fonster's extensions: Second Goppatoth · Gippatoth · Gappatoth · Geepatoth · Boppatoth · Troppatoth · Quadroppatoth · Humongulusplex · Tridecatrax · Tridecatreex · Tridecatrowx · Tridecatrux · Gonguldeus · Humingulus · Gongultreus · Humangulus · Gongulquadeus · Possible interpretation of tetrakulus · Humeengulus · Generatrix · Incredulus · Incridulus · Incradulus · Increedulus · Tercredulus · Tercridulus · Tercradulus · Tercreedulus · Pencredulus · Excredulus · Epcredulus · Ogcredulus · Goobawamba · Gibbawamba · Gabbawamba · Boobawamba · Troobawamba · Goobawantra · Emperatrix · Gossablossla · Hyperlatrix · Goobaduamba · Goobatriombia · Goobaquardimbia · Goobaquindingia · Goobasesixtia · Goobasebiptia · Goobagogdiktia · Goobanogdiktia · Xaplorgulus · Goobadektimtia · Goxxablorg · Gooxadworg · Goxxathrorg · Goxxaquorg · Cosslorgulus · Tesslorgulus · Pesslorgulus · Hesslorgulus · Zesslorgulus · Yosslorgulus · Brosslorgulus · Gosslorgulus · Mosslorgulus · Hasslorgulus · Kysslorgulus · Pisslorgulus · Sasslorgulus · Pexxlorgulus · Nisslorgulus · Zozzlorgulus · Ginglapulus · Ganglapulus · Geenglapulus · Bolapulus · Trolapulus · Goplapulus · Goduplapulus · Gotriplapulus · Hypertriakulus · Tetrakulus · Pentakulus · Hexakulus · Heptakulus · Oktakulus · Ennakulus · Big Mac · The Whopper
Douglas Shamlin Jr.'s extensions: Gongulquideus · Gongulsideus · Pencridulus · Geebawamba · Gibawamba · Gobbawamba · Gabawamba · Mulporawamba · Bibbawamba · Babbawamba · Beebawamba · Tribbawamba · Trabbawamba · Quadroobawamba · Quintoobawamba · Gibbawantra · Gabbawantra · Boobawantra · Troobawantra · Goobawanquadra · Goobawanquinta · Gissablossla · Gassablossla · Mossablossla · Missablossla · Massablossla · Bossablossla · Trossablossla · Quadrossablossla · Quintossablossla · Diteratrix · Gibbaduamba · Goobaduantra · Admiratrix · Dutridecatrix · Grand xaplorgulus · Binglapulus · Quadrolapulus · Giplapulus · Boplapulus · Goquadriplapulus · Goquintiplapulus · Golapulusduplex
BAT4423's extensions: Pencradulus · Pencreedulus · Encredulus · Powporawamba · Terporawamba · Bibawamba · Coxxablorg · Coxxadworg · Toxxablorg · Poxxablorg · Hexxablorg · Zexxablorg · Yoxxablorg · Broxxablorg · Gexxablorg · Cosmatrix · Alphlorgulus · Besslorgulus · Gasslorgulus · Desslorgulus · Thesslorgulus · Iottlorgulus · Kapplorgulus · Lasslorgulus · Sigglorgulus · Quintolapulus · Gosextiplapulus · Goseptiplapulus · Go-octiplapulus · Hectakulus · Chiliakulus · Megakulus · Gigakulus · Terakulus · Petakulus · Exakulus · Zettakulus · Yottakulus
MultiArrest91's extensions: Sextoobawamba · Geebawantra · Gibawantra · Gobbawantra · Gabawantra · Bibbawantra · Quadroobawantra · Boobawanquadra · Geesablossla · Gussablossla · Meesablossla · Mussablossla · Bissablossla · Bassablossla · Beesablossla · Sextossablossla · Goobaduanquadra · Triteratrix · Goobatriontria · Gaxxablorg · Geexablorg · Guxxablorg · Troxxablorg · Quadringlapulus · Sextolapulus · Biplapulus · Quadroplapulus · Giduplapulus · Boduplapulus
XRQ CORPORATION's extensions: Quintoppatoth · Octunculus · Mulporatrix · Humowngulus · Humungulus · Humagulus · Incrowdulus · Incrundulus · Tercrowdulus · Excridulus · Pepporawamba · Hexporawamba · Goobawansexta · Goomafiltra · Dutetradecatrix · Gixxablorg · Boxxablorg · Moxxablorg · Second Golapulus · Golapulusfact · Goxaplorglapulus · Gongcosslapulus · Gongtesslapulus · Gongpesslapulus · Ginglapulusfact · Gingxaplapulus · Gingcosslapulus · Gingtesslapulus · Gownglapulus · Gunglapulus · Banglapulus · Beenglapulus · Tringlapulus · Tranglapulus · Gaplapulus · Geeplapulus · Troplapulus · Quathragoth · Pentripluce · Quadrapulus · Tasspergulus · Sexanheldremhun · Septanheldremhun · Tridecadutrix · Humolapulus · Flastretchemorgulus · Elemongulus · Centikimcharcelkhandron · Bronquelmingaspegolapule · Lusspergulus · Firrpergulus · Josspergulus · Sosspergulus · Besspergulus · Gasspergulus · Supergulus · Verrpergulus · Multpergulus · Pysspergulus · Husspergulus · Icosakulus · Chick 'n Crisp · BBQ Bacon King · Troggpergulus · Basspergulus · Handpergulus · Quadruple Stacker King · Masspergulus · Gosspergulus · Whopper Jr. · Kosspergulus · Nesspergulus · Triple Whopper Jr. with Cheese · Western Whopper · Triple Whopper · Whopper Melt · Spicy Whopper Melt · Centuple Stacker King · Myriakulus · Xennakulus · Wekakulus · Vundakulus · Udakulus · Tredakulus · Sortakulus · Rintakulus · Quexakulus · Googolkulus · Googolplexikulus · The Whopperplex
RedBandServerYT's extensions: Kingulus · Kangulus · Keengulus · Bungulus · Trungulus · Quintunculus · Sextunculus · Septunculus · Dimendecatrix · The Whipper · The Whapper · The Wheeper · The Whowper · The Whupper
HaydenTheGoogologist2009's extensions: Goplapulusfact · Gogridplapulus · Gokubikplapulus · Gogolapulplapulus · Gogolapulplapulusfact · Gogolagridpulplapulus · Gogolakubikpulplapulus · Gogolapulusdeusplapulus · Gogolapulustruceplapulus · Gobolapulplapulus · Gotrolapulplapulus · Goplapulusdeus · Goplapulustruce · Hyper-golapulusfact · Hyper-ginglapulusfact · Hyper-ganglapulusfact · Hyper-bolapulusfact · Hyper-trolapulusfact · McRib · Bacon King · Single Stacker King · Double Quarter Pounder King · Double Stacker King · Triple Stacker King · BBQ Bacon Whopper · Bacon and Cheese Whopper · Double Whopper · Impossible Whopper
AuthorizeStructure's extensions: Dekulukulus · Dekulukulukulus
ARsygo's extensions: Gotopatoth · Goplexatoth · Kuplexulus · Second quadrunculus · Decredulus

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