Numbers from linear array notation

Here are some typical numbers from linear array notation. They’re also examples to help you understand LAN.

3-entry series

Those numbers are defined using s(a,b,c) with a, b, c > 1.

Tribo group

This group includes tribo, tetbo, pentbo and hexabo in order.

Tribo = s(3,3,2) = 3↑↑3 = 3^(3↑↑2) = 3^(3^(3↑↑1)) = 3^(3^3) = 3^27 = 7625597484987. It also equals those expressions: s(3,2,3) =(by rule 2) s(3,s(3,1,3),2) = s(3,s(3,1,2),2) = s(3,s(3,1,1),2) =(by rule 3) s(3,s(3,1),2) = s(3,3^1,2) = s(3,3,2). Using exAN, s(3,2{2}2) = s(3,2{1}1{1}2{2}1) = s(3,2,1,2) = s(3,3,2,1) = s(3,3,2). This numbers is quite small, even in real life. The amount of quarks in something weighing 32.2 grams, is about tribo squared. If we line tribo cubed quarks up, they just form a line about 47 thousand light years long, which is less than the radius of the milky way galaxy.

Tetbo = s(4,4,2) = 4↑↑4 = 4^(4↑↑3) = 4^(4^(4↑↑2)) = 4^(4^(4^(4↑↑1))) = 4^(4^(4^4)) = 4^(4^256) ≈ 10^(8.0723×10^153). It’s also equals those expressions: s(4,2,3) =(by rule 2) s(4,s(4,1,3),2) = s(4,s(4,1,1),2) =(by rule 3) s(4,s(4,1),2) = s(4,4^1,2) = s(4,4,2). Using exAN, s(4,2{2}2) = s(4,2{1}1{1}2{2}1) = s(4,2,1,2) = s(4,4,2,1) = s(4,4,2). This number gets large in real life. Tetbo Planck times is already longer than the Poincare recurrence time of the observed universe.

Pentbo = s(5,5,2) = 5↑↑5 = 5^(5↑↑4) = 5^(5^(5↑↑3)) = 5^(5^(5^(5↑↑2))) = 5^(5^(5^(5^(5↑↑1)))) = 5^(5^(5^(5^5))) = 5^(5^(5^3125)) ≈ 10^(10^(1.33574×10^2184)). It’s also equals those expressions: s(5,2,3) = s(5,s(5,1,3),2) = s(5,s(5,1,1),2) = s(5,s(5,1),2) = s(5,5^1,2) = s(5,5,2). Using exAN, s(5,2{2}2) = s(5,2{1}1{1}2{2}1) = s(5,2,1,2) = s(5,5,2,1) = s(5,5,2). Pentbo Planck times is longer than the Poincare recurrence time of a black hole with the same weight as the observed universe.

Hexabo = s(6,6,2) = 6↑↑6 = 6^(6↑↑5) = 6^(6^(6↑↑4)) = 6^(6^(6^(6↑↑3))) = 6^(6^(6^(6^(6↑↑2)))) = 6^(6^(6^(6^(6^(6↑↑1))))) = 6^(6^(6^(6^(6^6)))) = 6^(6^(6^(6^46656))) ≈ 10^(10^(10^(2.0692×10^36305))). It’s also equals those expressions: s(6,2,3) = s(6,s(6,1,3),2) = s(6,s(6,1,1),2) = s(6,s(6,1),2) = s(6,6^1,2) = s(6,6,2). Using exAN, s(6,2{2}2) = s(6,2{1}1{1}2{2}1) = s(6,2,1,2) = s(6,6,2,1) = s(6,6,2). This number stands above all the numbers used in real life science.

Trientri group

This group includes trientri, tettro, pentro and hextro in order.

Trientri = s(3,3,3) = 3↑↑↑3 = 3↑↑(3↑↑↑2) = 3↑↑(3↑↑(3↑↑↑1)) = 3↑↑(3↑↑3) = 3↑↑tribo = 3↑↑7625597484987. Also s(3,2,4) = s(3,s(3,1,4),3) = s(3,s(3,1,1),3) = s(3,s(3,1),3) = s(3,3^1,3) = s(3,3,3) = trientri, s(3,1,1,3) = s(3,3,1,2) = s(3,3,3,1) = s(3,3,3) = trientri, and s(3,1,1,1,2) = s(3,3,3,1,1) = s(3,3,3,1) = s(3,3,3) = trientri (so s(3,1,c,1,2) = trientri for any c). The name “trientri” comes from “3 entries of 3”. It’s also called tritri by Jonathan Bowers. Trientri can be expressed in exponentiation as follows:

Tettro = s(4,4,3) = 4↑↑↑4 = 4↑↑(4↑↑↑3) = 4↑↑(4↑↑(4↑↑↑2)) = 4↑↑(4↑↑(4↑↑(4↑↑↑1))) = 4↑↑(4↑↑(4↑↑4)) = 4↑↑(4↑↑tetbo). Also s(4,2,4) = s(4,s(4,1,4),3) = s(4,4,3) = tettro, and s(4,3,1,2) = s(4,4,3,1) = s(4,4,3) = tettro. Tettro can be expressed in exponentiation as follows:

Pentro = s(5,5,3) = 5↑↑↑5 = 5↑↑(5↑↑(5↑↑(5↑↑5))) = 5↑↑(5↑↑(5↑↑pentbo)). Also s(5,2,4) = s(5,s(5,1,4),3) = s(5,5,3) = pentro, and s(5,3,1,2) = s(5,5,3,1) = s(5,5,3) = pentro. Pentro can be expressed in exponentiation as follows:

Hextro = s(6,6,3) = 6↑↑↑6 = 6↑↑(6↑↑(6↑↑(6↑↑(6↑↑6)))) = 6↑↑(6↑↑(6↑↑(6↑↑hexabo))). Also s(6,2,4) = s(6,s(6,1,4),3) = s(6,6,3) = hextro, and s(6,3,1,2) = s(6,6,3,1) = s(6,6,3) = hextro. Hextro can be expressed in exponentiation as follows:

Trientet group

This group includes triteto, trientet, penteto and hexteto in order.

Triteto = s(3,3,4) = 3↑↑↑↑3 = 3↑↑↑(3↑↑↑3) = 3↑↑↑trientri. Also s(3,2,5) = s(3,s(3,1,5),4) = s(3,3,4) = triteto, and s(3,4,1,2) = s(3,3,4,1) = s(3,3,4) = triteto. It’s also called grahal by Aarex Tiaokhiao. Triteto can be expressed in exponentiation as follows:

Trientet = s(4,4,4) = 4↑↑↑↑4 = 4↑↑↑(4↑↑↑(4↑↑↑4)) = 4↑↑↑(4↑↑↑tettro). Also s(4,2,5) = s(4,s(4,1,5),4) = s(4,4,4) = trientet, s(4,1,1,3) = s(4,4,1,2) = s(4,4,4,1) = s(4,4,4) = trientet, and s(4,1,1,1,2) = s(4,4,4,1,1) = s(4,4,4,1) = s(4,4,4) = trientet. The name “trientet” comes from “3 entries of 4”. It’s also called tritet by Jonathan Bowers. Trientet can be expressed in exponentiation as follows:

Penteto = s(5,5,4) = 5↑↑↑↑5 = 5↑↑↑(5↑↑↑(5↑↑↑(5↑↑↑5))) = 5↑↑↑(5↑↑↑(5↑↑↑pentro)). Also s(5,2,5) = s(5,s(5,1,5),4) = s(5,5,4) = penteto, and s(5,4,1,2) = s(5,5,4,1) = s(5,5,4) = penteto. Penteto can be expressed in exponentiation as follows:

Hexteto = s(6,6,4) = 6↑↑↑↑6 = 6↑↑↑(6↑↑↑(6↑↑↑(6↑↑↑(6↑↑↑6)))) = 6↑↑↑(6↑↑↑(6↑↑↑(6↑↑↑hextro))). Also s(6,2,5) = s(6,s(6,1,5),4) = s(6,6,4) = hexteto, and s(6,4,1,2) = s(6,6,4,1) = s(6,6,4) = hexteto. Hexteto can be expressed in exponentiation as follows:

Trienpent group

This group includes tripeno, tetpeno, trienpent and hexpeno in order.

Tripeno = s(3,3,5) = 3↑↑↑↑↑3 = 3↑↑↑↑(3↑↑↑↑3) = 3↑↑↑↑triteto. Also s(3,2,6) = s(3,s(3,1,6),5) = s(3,3,5) = tripeno, and s(3,5,1,2) = s(3,3,5,1) = s(3,3,5) = tripeno. Tripeno can be expressed in exponentiation as follows:

Tetpeno = s(4,4,5) = 4↑↑↑↑↑4 = 4↑↑↑↑(4↑↑↑↑(4↑↑↑↑4)) = 4↑↑↑↑(4↑↑↑↑trientet). Also s(4,2,6) = s(4,s(4,1,6),5) = s(4,4,5) = tetpeno, and s(4,5,1,2) = s(4,4,5,1) = s(4,4,5) = tetpeno. Tetpeno can be expressed in exponentiation as follows:

Trienpent = s(5,5,5) = 5↑↑↑↑↑5 = 5↑↑↑↑(5↑↑↑↑(5↑↑↑↑(5↑↑↑↑5))) = 5↑↑↑↑(5↑↑↑↑(5↑↑↑↑penteto)). Also s(5,2,6) = s(5,s(5,1,6),5) = s(5,5,5) = trienpent, s(5,1,1,3) = s(5,5,1,2) = s(5,5,5,1) = s(5,5,5) = trienpent, and s(5,1,1,1,2) = s(5,5,5,1,1) = s(5,5,5,1) = s(5,5,5) = trienpent. It’s also called tripent by Jonathan Bowers. Trienpent can be expressed in exponentiation as follows:

Hexpeno = s(6,6,5) = 6↑↑↑↑↑6 = 6↑↑↑↑(6↑↑↑↑(6↑↑↑↑(6↑↑↑↑(6↑↑↑↑6)))) = 6↑↑↑↑(6↑↑↑↑(6↑↑↑↑(6↑↑↑↑hexteto))). Also s(6,2,6) = s(6,s(6,1,6),5) = s(6,6,5) = hexpeno, and s(6,5,1,2) = s(6,6,5,1) = s(6,6,5) = hexpeno. Hexpeno can be expressed in exponentiation as follows:

Trienhex group

This group includes trihexo, tethexo, penhexo and trienhex in order.

Trihexo = s(3,3,6) = 3↑↑↑↑↑↑3 = 3↑↑↑↑↑(3↑↑↑↑↑3) = 3↑↑↑↑↑tripeno. Also s(3,2,7) = s(3,s(3,1,7),6) = s(3,3,6) = trihexo, and s(3,6,1,2) = s(3,3,6,1) = s(3,3,6) = trihexo. Trihexo can be expressed in exponentiation as follows:

Tethexo, penhexo and trienhex are s(4,4,6), s(5,5,6) and s(6,6,6) respectively.

4-entry series

Those numbers are defined using s(a,b,c,d) with a, b, d > 1.

Primitol group

This group includes primitol, primitolplex, primitolbiplex, primitoltriplex, primitolquadriplex, primibol, primibolplex, primitrol, primitrolplex, primitetol, primitetolplex, primipenol and primipenolplex in order.

Primitol = s(3,2,2,2) = s(3,s(3,1,2,2),1,2) = s(3,s(3,1,1,2),1,2) = s(3,s(3,3,1,1),1,2) = s(3,s(3,3,1),1,2) = s(3,s(3,3),1,2) = s(3,3^3,1,2) = s(3,27,1,2) = s(3,3,27,1) = s(3,3,27) = 3↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑3. Let “stage 1” be 3^3 = 27, stage 2 = tribo = 3↑↑3 = 7625597484987, then stage 3 = trientri = 3↑↑↑3, stage 4 = triteto = 3↑↑↑↑3, stage 5 = tripeno = 3↑↑↑↑↑3 and stage 6 = trihexo = 3↑↑↑↑↑↑3. We have some pictures for stage 3 ~ 6, and you can image how large the stage 27 is – that’s primitol. It’s hard to define it in primitive recursive arithmetic, so it’s named “primitol”.

Primitolplex = s(3,3,2,2) = s(3,s(3,2,2,2),1,2) = s(3,3,s(3,2,2,2),1) = s(3,3,s(3,2,2,2)) = s(3,3,s(3,3,27)) = s(3,3,primitol). Also s(3,2,1,3) = s(3,3,2,2) = primitolplex. It’s the “stage primitol” number described above. Primitolplex can be expressed in up-arrow notation as follows:

Then, primitolbiplex, primitoltriplex and primitolquadriplex are s(3,4,2,2), s(3,5,2,2) and s(3,6,2,2) respectively. In up-arrow notation, they can be expressed as,andrespectively.

Primibol = s(3,2,3,2) = s(3,s(3,1,3,2),2,2) = s(3,s(3,1,2,2),2,2) = s(3,s(3,1,1,2),2,2) = s(3,s(3,3,1,1),2,2) = s(3,27,2,2). We can say it’s “primitol-25-plex”. Primibol can be expressed in up-arrow notation as follows:

Primibolplex = s(3,3,3,2) = s(3,s(3,2,3,2),2,2) = s(3,primibol,2,2). Also s(3,1,1,4) = s(3,3,1,3) = s(3,3,3,2) = primibolplex, and in exAN, s(3,2,1{2}2) = s(3,2,1,1,2{2}1) = s(3,2,1,1,2) = s(3,3,3,2,1) = s(3,3,3,2) = primibolplex. It also equals 3→3→3→3 in Conway’s chained arrow notation. Primibolplex can be expressed in up-arrow notation as follows:

Then, primitrol, primitetol and primipenol are s(3,2,4,2), s(3,2,5,2) and s(3,2,6,2) respectively. Primitrolplex, primitetolplex and primipenolplex are s(3,n,1,3) = s(3,3,n,2) with n = 4, 5 and 6 respectively.

Primitrol, primitrolplex, primitetol and primitetolplex can be expressed in up-arrow notation as,,andrespectively. (The pictures for primipenol and primipenolplex are too large and complex)

Tetentri group

This group includes duprimitol, duprimitolplex, duprimibol and tetentri in order.

Duprimitol = s(3,2,2,3) = s(3,s(3,1,2,3),1,3) = s(3,s(3,1,1,3),1,3) = s(3,s(3,3,1,2),1,3) = s(3,s(3,3,3,1),1,3) = s(3,s(3,3,3),1,3) = s(3,3,s(3,3,3),2) = s(3,3,trientri,2). We can say it’s “primi-(trientri-1)-olplex”. Then, duprimitolplex = s(3,3,2,3) = s(3,s(3,2,2,3),1,3) = s(3,3,s(3,2,2,3),2) = s(3,3,duprimitol,2), also s(3,2,1,4) = s(3,3,2,3) = duprimitolplex. And duprimibol = s(3,2,3,3) = s(3,s(3,1,3,3),2,3) = s(3,s(3,1,1,3),2,3) = s(3,trientri,2,3).

Tetentri = s(3,3,3,3) = s(3,s(3,2,3,3),2,3) = s(3,duprimibol,2,3). Also s(3,1,1,5) = s(3,3,1,4) = s(3,3,3,3) = tetentri, s(3,1,1,2,2) = s(3,3,1,1,2) = s(3,3,3,3,1) = s(3,3,3,3) = tetentri, s(3,1,1,1,1,2) = s(3,3,3,3,1,1) = s(3,3,3,3,1) = s(3,3,3,3) = tetentri, and in exAN s(3,3{2}2) = s(3,3,1,1,2{2}1) = s(3,3,1,1,2) = s(3,3,3,3,1) = s(3,3,3,3) = tetentri. It also equals 3→3→3→3→3 in Conway’s chained arrow notation. The name “tetentri” comes from “4 entries of 3”, but we can also name it duprimibolplex.

Tetentet group

This group includes truprimitol, truprimitolplex, truprimibol, truprimibolplex and tetentet in order.

Truprimitol = s(3,2,2,4) = s(3,s(3,1,2,4),1,4) = s(3,s(3,1,1,4),1,4) = s(3,s(3,3,1,3),1,4) = s(3,s(3,3,3,2),1,4) = s(3,3,s(3,3,3,2),3) = s(3,3,primibolplex,3). Truprimitolplex = s(3,3,2,4) = s(3,s(3,2,2,4),1,4) = s(3,3,s(3,2,2,4),3) = s(3,3,truprimitol,3), also s(3,2,1,5) = s(3,3,2,4) = truprimitolplex.

Truprimibol = s(3,2,3,4) = s(3,s(3,1,3,4),2,4) = s(3,s(3,1,1,4),2,4) = s(3,s(3,3,3,2),2,4) = s(3,primibolplex,2,4). Truprimibolplex = s(3,3,3,4) = s(3,s(3,2,3,4),2,4) = s(3,truprimibol,2,4), also s(3,1,1,6) = s(3,3,1,5) = s(3,3,3,4) = truprimibolplex, and s(3,4,1,1,2) = s(3,3,3,4,1) = s(3,3,3,4) = truprimibolplex.

Tetentet = s(4,4,4,4) = s(4,s(4,3,4,4),3,4) = s(4,s(4,s(4,2,4,4),3,4),3,4) = s(4,s(4,s(4,s(4,1,4,4),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,1,1,4),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,4,1,3),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,4,4,2),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,s(4,s(4,s(4,1,4,2),3,2),3,2),3,2),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,s(4,s(4,s(4,1,1,2),3,2),3,2),3,2),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,s(4,s(4,s(4,4,1,1),3,2),3,2),3,2),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,s(4,s(4,256,3,2),3,2),3,2),3,4),3,4),3,4). Also s(4,1,1,6) = s(4,4,1,5) = s(4,4,4,4) = tetentet, s(4,1,1,2,2) = s(4,4,1,1,2) = s(4,4,4,4,1) = s(4,4,4,4) = tetentet, and s(4,1,1,1,1,2) = s(4,4,4,4,1,1) = s(4,4,4,4,1) = s(4,4,4,4) = tetentet. It also equals 4→4→4→4→4→4 in Conway’s chained arrow notation.

Tetenpent group

This group includes quadprimitol, quadprimitolplex, quadprimibol, quadprimibolplex and tetenpent in order.

Quadprimitol = s(3,2,2,5) = s(3,s(3,1,2,5),1,5) = s(3,s(3,1,1,5),1,5) = s(3,s(3,3,1,4),1,5) = s(3,s(3,3,3,3),1,5) = s(3,3,s(3,3,3,3),4) = s(3,3,tetentri,4). Quadprimitolplex = s(3,3,2,5) = s(3,s(3,2,2,5),1,5) = s(3,3,s(3,2,2,5),4) = s(3,3,quadprimitol,4), also s(3,2,1,6) = s(3,3,2,5) = quadprimitolplex.

Quadprimibol = s(3,2,3,5) = s(3,s(3,1,3,5),2,5) = s(3,s(3,1,1,5),2,5) = s(3,tetentri,2,5). Quadprimibolplex = s(3,3,3,5) = s(3,s(3,2,3,5),2,5) = s(3,quadprimibol,2,5), also s(3,1,1,7) = s(3,3,1,6) = s(3,3,3,5) = quadprimibolplex, and s(3,5,1,1,2) = s(3,3,3,5,1) = s(3,3,3,5) = quadprimibolplex.

Tetenpent = s(5,5,5,5) = s(5,s(5,s(5,s(5,s(5,1,5,5),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,1,1,5),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,5,1,4),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,5,5,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,1,5,3),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,1,1,3),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,5,1,2),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,5,5,1),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,5,5),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,trienpent,4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5). Also s(5,1,1,7) = s(5,5,1,6) = s(5,5,5,5) = tetenpent, s(5,1,1,2,2) = s(5,5,1,1,2) = s(5,5,5,5,1) = s(5,5,5,5) = tetenpent, and s(5,1,1,1,1,2) = s(5,5,5,5,1,1) = s(5,5,5,5,1) = s(5,5,5,5) = tetenpent. It also equals 5→5→5→5→5→5→5 in Conway’s chained arrow notation.

Tetenhex group

This group includes quinprimitol, quinprimitolplex, quinprimibol, quinprimibolplex and tetenhex in order.

Quinprimitol = s(3,2,2,6) = s(3,s(3,1,2,6),1,6) = s(3,s(3,1,1,6),1,6) = s(3,s(3,3,1,5),1,6) = s(3,s(3,3,3,4),1,6) = s(3,3,s(3,3,3,4),5) = s(3,3,truprimibolplex,5). Quinprimitolplex = s(3,3,2,6) = s(3,s(3,2,2,6),1,6) = s(3,3,s(3,2,2,6),5) = s(3,3,quinprimitol,5), also s(3,2,1,7) = s(3,3,2,6) = quinprimitolplex.

Quinprimibol = s(3,2,3,6) = s(3,s(3,1,3,6),2,6) = s(3,s(3,1,1,6),2,6) = s(3,truprimibolplex,2,6). Quinprimibolplex = s(3,3,3,6) = s(3,s(3,2,3,6),2,6) = s(3,quinprimibol,2,6), also s(3,1,1,8) = s(3,3,1,7) = s(3,3,3,6) = quinprimibolplex, and s(3,6,1,1,2) = s(3,3,3,6,1) = s(3,3,3,6) = quinprimibolplex.

Tetenhex = s(6,6,6,6) = s(6,s(6,s(6,s(6,s(6,s(6,1,6,6),5,6),5,6),5,6),5,6),5,6), where s(6,1,6,6) = s(6,1,1,6) = s(6,6,1,5) = s(6,6,6,4) = s(6,s(6,s(6,s(6,s(6,s(6,1,6,4),5,4),5,4),5,4),5,4),5,4), where s(6,1,6,4) = s(6,1,1,4) = s(6,6,1,3) = s(6,6,6,2) = s(6,s(6,s(6,s(6,s(6,s(6,1,6,2),5,2),5,2),5,2),5,2),5,2), where s(6,1,6,2) = s(6,1,1,2) = s(6,6,1,1) = s(6,6,1) = s(6,6) = 46656. So tetenhex = s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,46656,5,2),5,2),5,2),5,2),5,2),5,4),5,4),5,4),5,4),5,4),5,6),5,6),5,6),5,6),5,6). Also s(6,1,1,8) = s(6,6,1,7) = s(6,6,6,6) = tetenhex, s(6,1,1,2,2) = s(6,6,1,1,2) = s(6,6,6,6,1) = s(6,6,6,6) = tetenhex, and s(6,1,1,1,1,2) = s(6,6,6,6,1,1) = s(6,6,6,6,1) = s(6,6,6,6) = tetenhex. It also equals 6→6→6→6→6→6→6→6 in Conway’s chained arrow notation.

5+ entry series

Those numbers are defined using LAN of 5 or more entries.

Chainol group

This group includes chainol, chainolplex, chainolbiplex, chainoltriplex, chainolquadriplex, chainbol, chainbolplex, chaintrol and chaintrolplex in order.

Chainol = s(3,2,2,1,2) = s(3,s(3,1,2,1,2),1,1,2) = s(3,s(3,1,1,1,2),1,1,2) = s(3,s(3,3,3,1,1),1,1,2) = s(3,s(3,3,3,1),1,1,2) = s(3,s(3,3,3),1,1,2) = s(3,3,3,s(3,3,3),1) = s(3,3,3,s(3,3,3)) = s(3,3,3,trientri) = 3→3→3→…3→3→3 with trientri+2 3′s. Let stage 1 = trientri, stage 2 = primibolplex, stage 3 = tetentri, stage 4 = truprimibolplex, stage 5 = quadprimibolplex and stage 6 = quinprimibolplex, etc. then chainol is the stage trientri number. It’s very hard to define (or bound) it in Conway’s chained arrow notation, so it’s named “chainol”.

Chainolplex = s(3,3,2,1,2) = s(3,s(3,2,2,1,2),1,1,2) = s(3,3,3,s(3,2,2,1,2),1) = s(3,3,3,s(3,2,2,1,2)) = s(3,3,3,chainol). Also s(3,2,1,2,2) = s(3,3,2,1,2) = chainolplex. Then chainolbiplex, chainoltriplex and chainolquadriplex are s(3,4,2,1,2), s(3,5,2,1,2) and s(3,6,2,1,2) respectively.

Chainbol = s(3,2,3,1,2) = s(3,s(3,1,3,1,2),2,1,2) = s(3,s(3,1,1,1,2),2,1,2) = s(3,trientri,2,1,2). Chainbolplex = s(3,3,3,1,2) = s(3,s(3,2,3,1,2),2,1,2) = s(3,chainbol,2,1,2), also s(3,1,1,3,2) = s(3,3,1,2,2) = s(3,3,3,1,2) = chainbolplex, and s(3,1,1,1,3) = s(3,3,3,1,2) = chainbolplex.

Chaintrol = s(3,2,4,1,2) = s(3,s(3,1,4,1,2),3,1,2) = s(3,s(3,1,1,1,2),3,1,2) = s(3,trientri,3,1,2). Chaintrolplex = s(3,3,4,1,2) = s(3,s(3,2,4,1,2),3,1,2) = s(3,chaintrol,3,1,2), also s(3,4,1,2,2) = s(3,3,4,1,2) = chaintrolplex.

Chainprimol group

This group includes chainprimol, chainprimolplex, chainpribol, chainpribolplex, chainduprimol, chainduprimolplex, chaindupribol, chaindupribolplex, chaintruprimol, chaintruprimolplex, chaintrupribol and chaintrupribolplex in order.

Chainprimol = s(3,2,2,2,2) = s(3,s(3,1,2,2,2),1,2,2) = s(3,s(3,1,1,2,2),1,2,2) = s(3,s(3,3,1,1,2),1,2,2) = s(3,s(3,3,3,3,1),1,2,2) = s(3,s(3,3,3,3),1,2,2) = s(3,3,s(3,3,3,3),1,2) = s(3,3,tetentri,1,2), and we can say it’s chain-(tetentri-1)-olplex. Chainprimolplex = s(3,3,2,2,2) = s(3,s(3,2,2,2,2),1,2,2) = s(3,3,s(3,2,2,2,2),1,2) = s(3,3,chainprimol,1,2), also s(3,2,1,3,2) = s(3,3,2,2,2) = chainprimolplex.

Chainpribol = s(3,2,3,2,2) = s(3,s(3,1,3,2,2),2,2,2) = s(3,s(3,1,1,2,2),2,2,2) = s(3,tetentri,2,2,2), and we can say it’s chainprimol-(tetentri-2)-plex. Chainpribolplex = s(3,3,3,2,2) = s(3,s(3,2,3,2,2),2,2,2) = s(3,chainpribol,2,2,2), also s(3,2,1,1,3) = s(3,3,3,2,2) = chainpribolplex, and s(3,1,1,4,2) = s(3,3,1,3,2) = s(3,3,3,2,2) = chainpribolplex.

Chainduprimol = s(3,2,2,3,2) = s(3,s(3,1,2,3,2),1,3,2) = s(3,s(3,1,1,3,2),1,3,2) = s(3,s(3,3,1,2,2),1,3,2) = s(3,s(3,3,3,1,2),1,3,2) = s(3,3,s(3,3,3,1,2),2,2) = s(3,3,chainbolplex,2,2), and we can say it’s chainpri-(chainbolplex-1)-olplex. Chainduprimolplex = s(3,3,2,3,2) = s(3,s(3,2,2,3,2),1,3,2) = s(3,3,s(3,2,2,3,2),2,2) = s(3,3,chainduprimol,2,2), also s(3,2,1,4,2) = s(3,3,2,3,2) = chainduprimolplex.

Chaindupribol = s(3,2,3,3,2) = s(3,s(3,1,3,3,2),2,3,2) = s(3,s(3,1,1,3,2),2,3,2) = s(3,chainbolplex,2,3,2). Chaindupribolplex = s(3,3,3,3,2) = s(3,s(3,2,3,3,2),2,3,2) = s(3,chaindupribol,2,3,2), also s(3,1,1,5,2) = s(3,3,1,4,2) = s(3,3,3,3,2) = chaindupribolplex, s(3,1,1,2,3) = s(3,3,1,1,3) = s(3,3,3,3,2) = chaindupribolplex, and s(3,2,1,1,1,2) = s(3,3,3,3,2,1) = s(3,3,3,3,2) = chaindupribolplex.

Chaintruprimol = s(3,2,2,4,2) = s(3,s(3,1,2,4,2),1,4,2) = s(3,s(3,1,1,4,2),1,4,2) = s(3,s(3,3,1,3,2),1,4,2) = s(3,s(3,3,3,2,2),1,4,2) = s(3,3,s(3,3,3,2,2),3,2) = s(3,3,chainpribolplex,3,2). Chaintruprimolplex = s(3,3,2,4,2) = s(3,s(3,2,2,4,2),1,4,2) = s(3,3,s(3,2,2,4,2),3,2) = s(3,3,chaintruprimol,3,2). Chaintrupribol = s(3,2,3,4,2) = s(3,s(3,1,3,4,2),2,4,2) = s(3,s(3,1,1,4,2),2,4,2) = s(3,chainpribolplex,2,4,2). Chaintrupribolplex = s(3,3,3,4,2) = s(3,s(3,2,3,4,2),2,4,2) = s(3,chaintrupribol,2,4,2), also s(3,4,1,1,3) = s(3,3,3,4,2) = chaintrupribolplex, and s(3,1,1,6,2) = s(3,3,1,5,2) = s(3,3,3,4,2) = chaintrupribolplex.

Pententri group

This group includes duchainol, duchainolplex, duchainbol, duchainbolplex, duchainprimol, duchainprimolplex, duchainpribol, duchainpribolplex, duchainduprimol, duchainduprimolplex, duchaindupribol and pententri in order.

Duchainol = s(3,2,2,1,3) = s(3,s(3,1,2,1,3),1,1,3) = s(3,s(3,1,1,1,3),1,1,3) = s(3,s(3,3,3,1,2),1,1,3) = s(3,3,3,s(3,3,3,1,2),2) = s(3,3,3,chainbolplex,2), and we can say it’s chain-(chainbolplex-1)-upribolplex. Duchainolplex = s(3,3,2,1,3) = s(3,s(3,2,2,1,3),1,1,3) = s(3,3,3,s(3,2,2,1,3),2) = s(3,3,3,duchainol,2), also s(3,2,1,2,3) = s(3,3,2,1,3) = duchainolplex. Duchainbol = s(3,2,3,1,3) = s(3,s(3,1,3,1,3),2,1,3) = s(3,s(3,1,1,1,3),2,1,3) = s(3,s(3,3,3,1,2),2,1,3) = s(3,chainbolplex,2,1,3). Duchainbolplex = s(3,3,3,1,3) = s(3,s(3,2,3,1,3),2,1,3) = s(3,duchainbol,2,1,3), also s(3,1,1,3,3) = s(3,3,1,2,3) = s(3,3,3,1,3) = duchainbolplex, and s(3,1,1,1,4) = s(3,3,3,1,3) = duchainbolplex.

Duchainprimol = s(3,2,2,2,3) = s(3,s(3,1,2,2,3),1,2,3) = s(3,s(3,1,1,2,3),1,2,3) = s(3,s(3,3,1,1,3),1,2,3) = s(3,s(3,3,3,3,2),1,2,3) = s(3,3,s(3,3,3,3,2),1,3) = s(3,3,chaindupribolplex,1,3), and we can say it’s duchain-(chaindupribolplex-1)-olplex. Duchainprimolplex = s(3,3,2,2,3) = s(3,s(3,2,2,2,3),1,2,3) = s(3,3,s(3,2,2,2,3),1,3) = s(3,3,duchainprimol,1,3). Duchainpribol = s(3,2,3,2,3) = s(3,s(3,1,3,2,3),2,2,3) = s(3,s(3,1,1,2,3),2,2,3) = s(3,chaindupribolplex,2,2,3). Duchainpribolplex = s(3,3,3,2,3) = s(3,s(3,2,3,2,3),2,2,3) = s(3,duchainpribol,2,2,3), also s(3,1,1,4,3) = s(3,3,1,3,3) = s(3,3,3,2,3) = duchainpribolplex, and s(3,2,1,1,4) = s(3,3,3,2,3) = duchainpribolplex.

Duchainduprimol = s(3,2,2,3,3) = s(3,s(3,1,2,3,3),1,3,3) = s(3,s(3,1,1,3,3),1,3,3) = s(3,s(3,3,1,2,3),1,3,3) = s(3,s(3,3,3,1,3),1,3,3) = s(3,3,duchainbolplex,2,3). Duchainduprimolplex = s(3,3,2,3,3) = s(3,s(3,2,2,3,3),1,3,3) = s(3,3,s(3,2,2,3,3),2,3) = s(3,3,duchainduprimol,2,3). Duchaindupribol = s(3,2,3,3,3) = s(3,s(3,1,3,3,3),2,3,3) = s(3,s(3,1,1,3,3),2,3,3) = s(3,duchainbolplex,2,3,3).

Pententri = s(3,3,3,3,3) = s(3,s(3,2,3,3,3),2,3,3) = s(3,duchaindupribol,2,3,3). Also s(3,1,1,5,3) = s(3,3,1,4,3) = s(3,3,3,3,3) = pententri, s(3,1,1,2,4) = s(3,3,1,1,4) = s(3,3,3,3,3) = pententri, s(3,1,1,2,1,2) = s(3,3,1,1,1,2) = s(3,3,3,3,3,1) = s(3,3,3,3,3) = pententri, s(3,1,1,1,1,1,2) = s(3,3,3,3,3,1,1) = s(3,3,3,3,3,1) = s(3,3,3,3,3) = pententri, and in exAN, s(3,3,1{2}2) = s(3,3,1,1,1,2{2}1) = s(3,3,1,1,1,2) = s(3,3,3,3,3,1) = s(3,3,3,3,3) = pententri. We can also name it duchaindupribolplex.

Pententet group

This group includes truchainol, truchainolplex, truchainbol, truchainbolplex, truchainprimol, truchainprimolplex, truchainpribol, truchainpribolplex, truchainduprimol, truchainduprimolplex, truchaindupribol, truchaindupribolplex, pententet, pentenpent and pentenhex in order.

Truchainol = s(3,2,2,1,4) = s(3,s(3,1,2,1,4),1,1,4) = s(3,s(3,1,1,1,4),1,1,4) = s(3,s(3,3,3,1,3),1,1,4) = s(3,3,3,s(3,3,3,1,3),3) = s(3,3,3,duchainbolplex,3). Truchainolplex = s(3,3,2,1,4) = s(3,s(3,2,2,1,4),1,1,4) = s(3,3,3,s(3,2,2,1,4),3) = s(3,3,3,truchainol,3). Truchainbol = s(3,2,3,1,4) = s(3,s(3,1,3,1,4),2,1,4) = s(3,s(3,1,1,1,4),2,1,4) = s(3,duchainbolplex,2,1,4).Truchainbolplex = s(3,3,3,1,4) = s(3,s(3,2,3,1,4),2,1,4) = s(3,truchainbol,2,1,4).

Truchainprimol = s(3,2,2,2,4) = s(3,s(3,1,2,2,4),1,2,4) = s(3,s(3,1,1,2,4),1,2,4) = s(3,s(3,3,1,1,4),1,2,4) = s(3,s(3,3,3,3,3),1,2,4) = s(3,3,s(3,3,3,3,3),1,4) = s(3,3,pententri,1,4). Truchainprimolplex = s(3,3,2,2,4) = s(3,s(3,2,2,2,4),1,2,4) = s(3,3,s(3,2,2,2,4),1,4) = s(3,3,truchainprimol,1,4). Truchainpribol = s(3,2,3,2,4) = s(3,s(3,1,3,2,4),2,2,4) = s(3,s(3,1,1,2,4),2,2,4) = s(3,pententri,2,2,4). Truchainpribolplex = s(3,3,3,2,4) = s(3,s(3,2,3,2,4),2,2,4) = s(3,truchainpribol,2,2,4).

Truchainduprimol = s(3,2,2,3,4) = s(3,s(3,1,2,3,4),1,3,4) = s(3,s(3,1,1,3,4),1,3,4) = s(3,s(3,3,1,2,4),1,3,4) = s(3,s(3,3,3,1,4),1,3,4) = s(3,3,s(3,3,3,1,4),2,4) = s(3,3,truchainbolplex,2,4). Truchainduprimolplex = s(3,3,2,3,4) = s(3,s(3,2,2,3,4),1,3,4) = s(3,3,s(3,2,2,3,4),2,4) = s(3,3,truchainduprimol,2,4). Truchaindupribol = s(3,2,3,3,4) = s(3,s(3,1,3,3,4),2,3,4) = s(3,s(3,1,1,3,4),2,3,4) = s(3,truchainbolplex,2,3,4). Truchaindupribolplex = s(3,3,3,3,4) = s(3,s(3,2,3,3,4),2,3,4) = s(3,truchaindupribol,2,3,4), also s(3,1,1,5,4) = s(3,3,1,4,4) = s(3,3,3,3,4) = truchaindupribolplex, s(3,1,1,2,5) = s(3,3,1,1,5) = s(3,3,3,3,4) = truchaindupribolplex, and in exAN, s(3,4{2}2) = s(3,4,1,1,1,2{2}1) = s(3,4,1,1,1,2) = s(3,3,3,3,4,1) = s(3,3,3,3,4) = truchaindupribolplex.

And, pententet, pentenpent and pentenhex are s(4,4,4,4,4), s(5,5,5,5,5) and s(6,6,6,6,6) respectively. Especially, in exAN, s(4,4{2}2) = s(4,4,1,1,1,2{2}1) = s(4,4,1,1,1,2) = s(4,4,4,4,4,1) = s(4,4,4,4,4) = pententet.

Choinol group

This group includes choinol, choinalplex, choinalbiplex, choinbol, choinbolplex, choinprimol, choinprimolplex, choinpribol, choinpribolplex, choinduprimol, choinduprimolplex, choindupribol and choindupribolplex in order.

Choinol = s(3,2,2,1,1,2) = s(3,s(3,1,2,1,1,2),1,1,1,2) = s(3,s(3,1,1,1,1,2),1,1,1,2) = s(3,s(3,3,3,3,1,1),1,1,1,2) = s(3,s(3,3,3,3),1,1,1,2) = s(3,3,3,3,s(3,3,3,3),1) = s(3,3,3,3,s(3,3,3,3)) = s(3,3,3,3,tetentri), and we can say it’s (tetentri-1)-uchaindupribolplex.

Choinolplex = s(3,3,2,1,1,2) = s(3,s(3,2,2,1,1,2),1,1,1,2) = s(3,3,3,3,s(3,2,2,1,1,2),1) = s(3,3,3,3,s(3,2,2,1,1,2)) = s(3,3,3,3,choinol), and choinalbiplex = s(3,4,2,1,1,2) = s(3,3,3,3,choinalplex).

Choinbol = s(3,2,3,1,1,2) = s(3,s(3,1,3,1,1,2),2,1,1,2) = s(3,s(3,1,1,1,1,2),2,1,1,2) = s(3,tetentri,2,1,1,2). Choinbolplex = s(3,3,3,1,1,2) = s(3,s(3,2,3,1,1,2),2,1,1,2) = s(3,choinbol,2,1,1,2), also s(3,1,1,3,1,2) = s(3,3,1,2,1,2) = s(3,3,3,1,1,2) = choinbolplex, and s(3,1,1,1,2,2) = s(3,3,3,1,1,2) = choinbolplex.

Choinprimol = s(3,2,2,2,1,2) = s(3,s(3,1,2,2,1,2),1,2,1,2) = s(3,s(3,1,1,2,1,2),1,2,1,2) = s(3,s(3,3,1,1,1,2),1,2,1,2) = s(3,s(3,3,3,3,3,1),1,2,1,2) = s(3,s(3,3,3,3,3),1,2,1,2) = s(3,3,s(3,3,3,3,3),1,1,2) = s(3,3,pententri,1,1,2). Choinprimolplex = s(3,3,2,2,1,2) = s(3,s(3,2,2,2,1,2),1,2,1,2) = s(3,3,s(3,2,2,2,1,2),1,1,2) = s(3,3,choinprimol,1,1,2). Choinpribol = s(3,2,3,2,1,2) = s(3,s(3,1,3,2,1,2),2,2,1,2) = s(3,s(3,1,1,2,1,2),2,2,1,2) = s(3,pententri,2,2,1,2). Choinpribolplex = s(3,3,3,2,1,2) = s(3,s(3,2,3,2,1,2),2,2,1,2) = s(3,choinpribol,2,2,1,2).

Choinduprimol = s(3,2,2,3,1,2), choinduprimolplex = s(3,3,2,3,1,2), choindupribol = s(3,2,3,3,1,2), and choindupribolplex = s(3,3,3,3,1,2). Also s(3,1,1,5,1,2) = s(3,3,1,4,1,2) = s(3,3,3,3,1,2) = choindupribolplex, and s(3,1,1,2,2,2) = s(3,3,1,1,2,2) = s(3,3,3,3,1,2) = choindupribolplex.

Choichainol group

This group includes choichainol, choichainolplex, choichainbolplex, choichainpribolplex, choichaindupribolplex and choiduchaindupribolplex in order.

Choichainol = s(3,2,2,1,2,2) = s(3,s(3,1,2,1,2,2),1,1,2,2) = s(3,s(3,1,1,1,2,2),1,1,2,2) = s(3,s(3,3,3,1,1,2),1,1,2,2) = s(3,3,3,s(3,3,3,1,1,2),1,2) = s(3,3,3,choinbolplex,1,2). Choichainolplex = s(3,3,2,1,2,2) = s(3,s(3,2,2,1,2,2),1,1,2,2) = s(3,3,3,s(3,2,2,1,2,2),1,2) = s(3,3,3,choichainol,1,2). Choichainbolplex = s(3,3,3,1,2,2) = s(3,s(3,2,3,1,2,2),2,1,2,2) = s(3,s(3,s(3,1,3,1,2,2),2,1,2,2),2,1,2,2) = s(3,s(3,s(3,1,1,1,2,2),2,1,2,2),2,1,2,2) = s(3,s(3,choinbolplex,2,1,2,2),2,1,2,2), also s(3,1,1,3,2,2) = s(3,3,1,2,2,2) = s(3,3,3,1,2,2) = choichainbolplex.

Choichainpribolplex = s(3,3,3,2,2,2) = s(3,s(3,s(3,1,3,2,2,2),2,2,2,2),2,2,2,2) = s(3,s(3,s(3,1,1,2,2,2),2,2,2,2),2,2,2,2) = s(3,s(3,choindupribolplex,2,2,2,2),2,2,2,2). Choichaindupribolplex = s(3,3,3,3,2,2) = s(3,s(3,s(3,1,3,3,2,2),2,3,2,2),2,3,2,2) = s(3,s(3,s(3,1,1,3,2,2),2,3,2,2),2,3,2,2) = s(3,s(3,s(3,3,1,2,2,2),2,3,2,2),2,3,2,2) = s(3,s(3,s(3,3,3,1,2,2),2,3,2,2),2,3,2,2) = s(3,s(3,choichainbolplex,2,3,2,2),2,3,2,2). Also s(3,1,1,5,2,2) = s(3,3,1,4,2,2) = s(3,3,3,3,2,2) = choichaindupribolplex, and s(3,1,1,2,3,2) = s(3,3,1,1,3,2) = s(3,3,3,3,2,2) = choichaindupribolplex.

Choiduchaindupribolplex = s(3,3,3,3,3,2) = s(3,s(3,s(3,1,3,3,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,1,1,3,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,3,1,2,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,3,3,1,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,s(3,s(3,1,3,1,3,2),2,1,3,2),2,1,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,s(3,s(3,1,1,1,3,2),2,1,3,2),2,1,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,s(3,s(3,3,3,1,2,2),2,1,3,2),2,1,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,s(3,choichainbolplex,2,1,3,2),2,1,3,2),2,3,3,2),2,3,3,2). Also, s(3,1,1,5,3,2) = s(3,3,1,4,3,2) = s(3,3,3,3,3,2) = choiduchaindupribolplex, s(3,1,1,2,4,2) = s(3,3,1,1,4,2) = s(3,3,3,3,3,2) = choiduchaindupribolplex, and s(3,1,1,2,1,3) = s(3,3,1,1,1,3) = s(3,3,3,3,3,2) = choiduchaindupribolplex.

Up to here, the naming system works approximately as follows: e-uchoi-d-uchain-c-upri-b-ol-a-plex = s(3,a+2,b+1,c+1,d+1,e+1), where a, c, d, e ≥ 0 and b ≥ 1.

Hexentri group

Here’s the final group of LAN numbers. This group includes hexentri, hexentet, hexenpent and hexenhex.

Hexentri, hexentet, hexenpent and hexenhex are s(n,n,n,n,n,n) = s(n,n,1,1,1,1,2) = s(n,1,1,1,1,1,1,2) with n = 3, 4, 5 and 6 respectively. Especially, s(4,4,1{2}2) = s(4,4,1,1,1,1,2{2}1) = s(4,4,1,1,1,1,2) = hexentet, and s(5,5{2}2) = s(5,5,1,1,1,1,2{2}1) = s(5,5,1,1,1,1,2) = hexenpent.