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In Nov 2013 ~ Jan 2014, I first introduce R function series to you. In recent months, I realized that R function is still too complex and very hard to make extension. So I invented another recursion googological function.

The strong array notation is published here, including 11 parts currently.

An overview on my array notation

Linear array notation (LAN) is similar to the linear array notation in BEAF and BAN. It has the same growth rate but different definition. We use s(a,b,c,...,y,z) to present it.

Extended array notation (exAN) is similar to tetrational array notation in BEAF and nested array notation in BAN. The comma is a shorthand for {1} here.

Up to multiple expanding array notation (mEAN), the array notation is still very similar to BAN in growth rate. The m-ple grave accent (i.e. ```...`` with m grave accents) corresponds to the /m in BAN. But the definition is different. The definition is more similar to HAN.

In primary dropping array notation (pDAN), the double comma (,,) corresponds to the comma in original R function, and the { ____ ,,} separators correspond to { ____ *} in original R function.

In secondary dropping array notation (sDAN), the {1,,,2} corresponds to the comma (i.e. {0*}) in original R function, the {1,,,3} corresponds to {0**} in R function, the {1,,,4} corresponds to {0***} in R function, and so on. The triple comma corresponds to the double comma in original R function.

Dropping array notation (DAN) is comparable to the full original R function, though the separators are not fully the same. The m-ple comma (m > 1) in DAN corresponds to the (m-1)-ple comma in R function.

In array notation I cancelled the definition of "braces", and there're just separators. Without "braces not separators" the rules can be simplified a lot.

Nested dropper array notation (NDAN) is the first part where the array notation surpasses R function. From here on, we use 4 subfunctions to help to solve the separators. NDAN is a gap for my array notation, because the definition changes so much that some expressions result different value comparing to before.

Remaining work

The array notation can be extended further. Ideas for extensions are easy, but definitions for extensions are more difficult. To make the notation strong but simple, we need even more work.

Comparisons between my array notation and other notations are still incomplete. The strongest notation so far I know is Taranovsky's notation - an ordinal notation, and it may be the only notation that surpasses my array notation, but it's not defined by recursion. So it's hard work to make the comparisons.

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