Crescent notation

Crescent notation is an array notation defined by wiki user MiTerSkyArk, similar to other array notations.

Definition

• ☾a☽ = a
• ☾a, b☽ = ☾a, b, 1☽ = a^b
• ☾a, b, 2☽ = ☾a, ☾a, ☾a...☾a, a☽...☽☽☽ with ☾b, b☽ a's
• ☾a, b, 3☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 2☽, 2☽...☽, 2☽, 2☽ with ☾b, b, 2☽ a's
• ☾a, b, c☽ = ☾a, ☾a, ☾...☾a, ☾a, a c-1☽, c-1☽...☽, c-1☽, c-1☽ with ☾b, b, c-1☽ a's/recursions
• ☾a, b, 1, 2☽ = ☾a, a, ☾a, a, ☾...☾a, a, ☾a, a, a☽☽...☽☽☽ with ☾b, b, b☽ recursions
• ☾a, b, 2, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 2☽, 1, 2☽, 1, 2...☽, 1, 2☽, 1, 2☽ with ☾b, b, 1, 2☽ recursions
• ☾a, b, 3, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 2, 2☽, 2, 2☽, 2, 2...☽, 2, 2☽, 2, 2☽ with ☾b, b, 2, 2☽ recursions
• ☾a, b, c, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 2☽, c-1, 2☽, c-1, 2...☽, c-1, 2☽, c-1, 2☽ with ☾b, b, c-1, 2☽ recursions
• ☾a, b, 1, 3☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 2☽, 1, 2☽, 1, 2...☽, 1, 2☽, 1, 2☽ with ☾b, b, 1, 2☽ recursions
• ☾a, b, 2, 3☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 3☽, 1, 3☽, 1, 3...☽, 1, 3☽, 1, 3☽ with ☾b, b, 1, 3☽ recursions
• ☾a, b, c, 3☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 3☽, c-1, 3☽, c-1, 3...☽, c-1, 3☽, c-1, 3☽ with ☾b, b, c-1, 3☽ recursions
• ☾a, b, c, d☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, d☽, c-1, d☽, c-1, d...☽, c-1, d☽, c-1, d☽ with ☾b, b, c-1, d☽ recursions
• ☾a, b, 1, 1, 2☽ = ☾a, a, a, ☾a, a, a, ☾...☾a, a, a, ☾a, a, a, a☽☽...☽☽☽ with ☾b, b, b, b☽ recursions
• ☾a, b, 2, 1, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 1, 2☽, 1, 1, 2☽...☽, 1, 1, 2☽, 1, 1, 2☽ with ☾b, b, 1, 1, 2☽ recursions
• ☾a, b, c, 1, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 1, 2☽ c-1, 1, 2☽...☽, c-1, 1, 2☽, c-1, 1, 2☽ with ☾b, b, c-1, 1, 2☽ recursions
• ☾a, b, 1, 2, 2☽ = ☾a, a, ☾a, a, ☾...☾a, a, ☾a, a, 1, 1, 2☽, 1, 2☽...☽ 1, 2☽, 1, 2☽ with ☾b, b, 1, 1, 2☽ recursions
• ☾a, b, 2, 2, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 2, 2☽, 1, 2, 2☽...☽, 1, 2, 2☽, 1, 2, 2☽ with ☾b, b, 1, 2, 2☽ recursions
• ☾a, b, c, 2, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 2, 2☽, c-1, 2, 2☽...☽, c-1, 2, 2☽, c-1, 2, 2☽ with ☾b, b, c-1, 2, 2☽ recursions
• ☾a, b, 1, 3, 2☽ = ☾a, a, ☾a, a, ☾...☾a, a, ☾a, a, 1, 2, 2☽, 2, 2☽...☽, 2, 2☽, 2, 2☽ with ☾b, b, 1, 2, 2☽ recursions
• ☾a, b, 2, 3, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 3, 2☽, 1, 3, 2☽...☽, 1, 3, 2☽ 1, 3, 2☽ with ☾b, b, 1, 2, 2☽ recursions
• ☾a, b, c, 3, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 3, 2☽, c-1, 3, 2☽...☽, c-1, 3, 2☽, c-1, 3, 2☽ with ☾b, b, c-1, 3, 2☽ recursions
• ☾a, b, c, d, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, d, 2☽, c-1, d, 2☽...☽ c-1, d, 2☽, c-1, d, 2☽ with ☾b, b, c-1, d, 2☽ recursions
• ☾a, b, c, d, e☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, d, e☽, c-1, d, e☽...☽ c-1, d, e☽, c-1, d, e☽ with ☾b, b, c-1, d, e☽ recursions
• ☾a, b, 1, 1, 1, 2☽ = ☾a, a, a, a, ☾a, a, a, a, ☾...☾a, a, a, a, ☾a, a, a, a, a☽☽...☽☽☽ with ☾b, b, b, b, b☽ recursions
• ☾a, b, 2, 1, 1, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 1, 1, 2☽, 1, 1, 1, 2☽...☽ 1, 1, 1, 2☽, 1, 1, 1, 2☽ with ☾b, b, 1, 1, 1, 2☽ recursions
• ☾a, b, c, d, e, f☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, d, e, f☽, c-1, d, e, f☽...☽ c-1, d, e, f☽, c-1, d, e, f☽ with ☾b, b, c-1, d, e, f☽ recursions

and so on, with any number of entries.

• ☾a|2☽ = ☾a, a, a...a, a, a☽ with a entries of a
• ☾a|3☽ = ☾☾☾...☾a|2☽|2...☽|2☽|2☽ with ☾a|2☽ recursions
• ☾a|b☽ = ☾☾☾...☾a|b-1☽|b-1...☽|b-1☽|b-1☽ with ☾a|b-1☽ recursions
• ☾a|1, 2☽ = ☾☾☾...☾☾a|a☽|a☽...☽|a☽|a☽ with ☾a|a☽ a's
• ☾a|2, 2☽ = ☾☾☾...☾☾a|1, 2☽|1, 2☽...☽|1, 2☽|1, 2☽ with ☾a|1,2☽ ☾...|1,2☽'s
• ☾a|b, 2☽ = ☾☾☾...☾☾a|b-1, 2☽|b-1, 2☽...☽|b-1, 2☽|b-1, 2☽ with ☾a|b-1,2☽ ☾...|b-1,2☽'s
• ☾a|b, c☽ = ☾☾☾...☾☾a|b-1, c☽|b-1, c☽...☽|b-1, c☽|b-1, c☽ with ☾a|b-1, c☽ ☾...|\b-1,c☽'s
• ☾a|1, 1, 2☽ = ☾☾☾...☾☾a|a, a☽|a, a☽...☽|a, a☽|a, a☽ with ☾a|a, a☽ recursions
• ☾a|b, c, d☽ = ☾☾☾...☾☾a|b-1, c, d☽|b-1, c, d☽...☽|b-1, c, d☽|b-1, c, d☽ with ☾a|b-1, c, d☽ recursions

and so on, with any number of entries.

• ☾a||2☽ = ☾a|a, a, a...a, a☽ with a entries of a
• ☾a||b☽ = ☾☾☾....☾☾a||b-1☽||b-1☽...☽||b-1☽||b-1☽ with ☾a||b-1☽ recursions
• ☾a||1, 2☽ = ☾☾☾...☾☾a||a☽||a☽...☽||a☽||a☽ with ☾a||a☽ recursions
• ☾a||b, 2☽ = ☾☾☾....☾☾a||b-1, 2☽||b-1, 2☽...☽||b-1, 2☽||b-1, 2☽ with ☾a||b-1, 2☽ recursions
• ☾a||b, c☽ = ☾☾☾...☾☾a||b-1, c☽||b-1, c☽...☽||b-1, c☽||b-1, c☽ with ☾a||b-1, c☽ recursions
• ☾a||1, 1, 2☽ = ☾☾☾...☾☾a||a, a☽||a, a☽...☽||a, a☽||a, a☽ with ☾a||a, a☽ recursions
• ☾a||b, c, d☽ = ☾☾☾...☾☾a||b-1, c, d☽||b-1, c, d☽...☽||b-1, c, d☽||b-1, c, d☽ with ☾a||b-1, c, d☽ recursions

and so on, with any number of entries.

☾a|||b...☽ works in the same way as ☾a||b...☽ but with three |'s instead of two. And so on, with any number of separators.

• ☾a|1|2☽ = ☾a||...||a☽ with a separators = ☾a/2☽
• ☾a|2|2☽ = ☾☾☾...☾☾a|1|2☽|1|2☽...☽|1|2☽|1|2☽ with ☾a|1|2☽ recursions
• ☾a|b|2☽ = ☾☾☾...☾☾a|b-1|2☽|b-1|2☽...☽|b-1|2☽|b-1|2☽ with ☾a|b-1|2☽ recursions = ☾a/b☽
• ☾a|b|c☽ = ☾☾☾...☾☾a|b-1|c☽|b-1|c☽...☽|b-1|c☽|b-1|c☽ with ☾a|b-1|c☽ recursions
• ☾a|1, 1|2☽ ☾a|a|☾a|a|☾...☾a|a|☾a|a|a☽☽...☽☽☽ with ☾a|a|a☽ recursions
• ☾a|2, 1|2☽ = ☾☾☾...☾☾a|1, 1|2☽|1, 1|2☽...☽|1, 1|2☽|1, 1|2☽ with ☾a|1, 1|2☽ recursions
• ☾a|b, 1|2☽ = ☾☾☾...☾☾a|b-1, 1|2☽|b-1, 1, 2☽...☽|b-1, 1, 2☽|b-1, 1|2☽
• ☾a|1, 2|2☽ = ☾☾☾...☾☾a|a, 1|2☽|a, 1, 2☽...☽|a, 1|2☽|a, 1|2☽ with ☾a|a, 1|2☽ recursions
• ☾a|2, 2|2☽ = ☾☾☾...☾☾a|1, 2|2☽|1, 2|2☽...☽|1, 2|2☽|1, 2|2☽ with ☾a|1, 2|2☽ recursions
• ☾a|b, 2|2☽ = ☾☾☾...☾☾a|b-1,2|2☽|b-1, 2|2☽...☽|b-1, 2|2☽|b-1, 2|2☽ with ☾a|b-1,2|2☽ recursions
• ☾a|b, c|2☽ = ☾☾☾...☾☾a|b-1,c|2☽|b-1, c|2☽...☽|b-1, c|2☽|b-1, c|2☽ with ☾a|b-1, c|2☽ recursions
• ☾a|b, c|d☽ = ☾☾☾...☾☾a|b-1, c|d☽|b-1, c|d☽...☽|b-1, c|d☽|b-1, c|d☽ with ☾a|b-1, c|d☽ recursions
• ☾a|b, c, d|e☽ = ☾☾☾...☾☾a|b-1, c, d|e☽|b-1, c, d|e☽...☽|b-1, c, d|e☽|b-1, c, d|e☽ with ☾a|b-1, c, d|e☽ recursions

and so on, with any number of entries after the seperator

• ☾a|1|1, 2☽ = ☾a|a, a, a...a, a, a|a☽ with a entries of a in the middle
• ☾a|2|1, 2☽ = ☾☾☾...☾☾a|1|1, 2☽|1|1, 2☽...☽|1|1, 2☽|1|1, 2☽ with ☾a|1|1, 2☽ recursions
• ☾a|b|1, 2☽ = ☾☾☾...☾☾a|b-1|1, 2☽|b-1|1, 2☽...☽|b-1|1, 2☽|b-1|1, 2☽ with ☾a|b-1|1, 2☽ recursions
• ☾a|1|2, 2☽ = ☾☾☾...☾☾a|a|1, 2☽|a|1, 2☽...☽|a|1, 2☽|a|1, 2☽ with ☾a|a|1, 2☽ recursions
• ☾a|2|2, 2☽ = ☾☾☾...☾☾a|1|2, 2☽|1|2, 2☽...☽|1|2, 2☽|1|2, 2☽ with ☾a|1|2, 2☽ recursions
• ☾a|b|2, 2☽ = ☾☾☾...☾☾a|b-1|2, 2☽|b-1|2, 2☽...☽|b-1|2, 2☽|b-1|2, 2☽ with ☾a|b-1|2, 2☽ recursions
• ☾a|b|c, d☽ = ☾☾☾....☾☾a|b-1|c, d☽|b-1|c, d☽...☽|b-1|c, d☽|b-1|c, d☽ with ☾a|b-1|c, d☽ recursions
• ☾a|1|1, 1, 2☽ = ☾☾☾...☾☾a|a|a, a☽|a|a, a☽...☽|a|a, a☽|a|a, a☽ with ☾a|a|a, a☽ recursions
• ☾a|2|1, 1, 2☽ = ☾☾☾...☾☾a|1|1, 1, 2☽|1|1, 1, 2☽...☽|1|1, 1, 2☽|1|1, 1, 2☽ with ☾a|1|1, 1, 2☽ recursions
• ☾a|b|c, d, e☽ = ☾☾☾...☾☾a|b-1|c, d, e☽|b-1|c, d, e☽...☽|b-1|c, d, e☽|b-1|c, d, e☽ with ☾a|b-1|c, d, e☽ recursions

and so on, win any number of entries after the separator.

• ☾a|b...|c...☽ works with any number of entries in each. ☾a|b...|c...|d...|e...☽ any number of separators with any number of entries.

It also works with ☾a||b...||c...||d...☽ which works the same as with one separator between each, except with two.

• ☾a||1, 1||2☽ = ☾a|a, a, a...a, a, a|a, a, a...a, a, a|a, a, a......a, a, a☽ with a separators, and a a's between each.

this continues on with ☾a|||b...|||c....|||d...☽ and with 4, 5, and any number of separators with any number of entries between them.

• ☾a:2☽ = ☾a||...||a, a...a, a||...||a, a...a, a||...||......a, a||...||a, a...a, a☽ with a groups of a separators, with a a's between each group.
• ☾a:b☽ = ☾☾☾...☾☾a:b-1☽:b-1☽...☽:b-1☽:b-1☽ with ☾a:b-1☽ recursions.
• ☾a:b:2☽ = ☾a:☾a:☾...☾a:☾a:a☽☽...☽☽☽ with ☾b:b☽ recursions
• ☾a:b:c☽ = ☾a:☾a:☾...☾a:☾a:a:c-1☽:c-1☽...☽:c-1☽:c-1☽ with ☾b:b:c-1☽ recursions
• ☾a:b:1:2☽ = ☾a:a:☾a:a:☾...☾a:a:☾a:a:a☽☽...☽☽☽ with ☾b:b:b☽ recursions
• ☾a:b:2:2☽ = ☾a:☾a:☾...☾a:☾a:a:1:2☽:1:2☽...☽:1:2☽:1:2☽ with ☾a:a:1:2☽ recursions
• ☾a:b:c:2☽ = ☾a:☾a:☾...☾a:☾a:a:c-1:2☽:c-1:2☽...☽:c-1:2☽:c-1:2☽ with ☾a:a:c-1:2☽ recursions
• ☾a:b:1:3☽ = ☾a:a:☾a:a:☾...☾a:a:☾a:a:a:2☽:2☽...☽:2☽:2☽ with ☾a:a:a:2☽ recursions
• ☾a:b:c:d☽ = ☾a:☾a:☾...☾a:☾a:a:c-1:d☽:c-1:d☽...☽:c-1:d☽:c-1:d☽ with ☾a:a:c-1:d☽ recursions
• ☾a:b:1:1:2☽ = ☾a:a:a:☾a:a:a:☾...☾a:a:a:☾a:a:a:a☽☽...☽☽☽ with ☾a:a:a:a☽ recursions
• ☾a:b:c:d:e☽ = ☾a:☾a:☾...☾a:☾a:a:c-1:d:e☽:c-1:d:e☽...☽:c-1:d:e☽:c-1:d:e☽ with ☾a:a:c-1:d:e☽ recursions

and so on, with any number of entries.

• ☾a::2☽ = ☾a:a:a...a:a:a☽ with a a's
• ☾a::b☽ = ☾☾☾...☾☾a::b-1☽:b-1☽...☽:b-1☽:b-1☽ with ☾a::b-1☽ recursions

This continues in the same way as ☾a:b...☽ but with two :'s instead of one.

It continues in the same way with ☾a:::b☽, ☾a::::b☽, and so on.

• ☾a%2☽ = ☾a:::...:::a☽ with a :'s
• ☾a%b☽ = ☾☾☾...☾☾a%b-1☽%b-1☽...☽%b-1☽%b-1☽ with ☾a%b-1☽ recursions

☾a%b...☽ continues the same way as ☾a:b☽, except with % instead of :

• ☾a?☽ = ☾a%a-1%a-2...%4%3%2☽
• ☾a?2☽ = ☾☾☾...☾☾a?☽?☽...☽?☽?☽ with a ?'s
• ☾a?b☽ = ☾☾☾...☾☾a?b-1☽?b-1☽...☽?b-1☽?b-1☽ with ☾a?b-1☽ recursions

Approximations

• ☾n, n☽ ~ \(f_3(n)\)
• ☾n, n, 2☽ ~ \(f_4(n)\)
• ☾n, n, 3☽ ~ \(f_5(n)\)
• ☾n, n, n☽ ~ \(f_{\omega}(n)\)
• ☾n, n, 1, 2☽ ~ \(f_{\omega+1}(n)\)
• ☾n, n, 2, 2☽ ~ \(f_{\omega+2}(n)\)
• ☾n, n, n, 2☽ ~ \(f_{\omega 2}(n)\)
• ☾n, n, n, 3☽ ~ \(f_{\omega 3}(n)\)
• ☾n, n, n, n☽ ~ \(f_{\omega^2}(n)\)
• ☾n, n, n, n, 2☽ ~ \(_{\omega^2 2}(n)\)
• ☾n, n, n, n, n☽ ~ \(_{\omega^3}(n)\)
• ☾n, n, n, n, n, n☽ ~ \(_{\omega^4}(n)\)
• ☾n|n☽ ~ \(_{\omega^{\omega}}(n)\)

• please continue the approximations. Current ones may be incorrect