In The Cambridge Grammar of the English Language, the word another is described on page 391:
Determinative another derives historically from the compounding of the indefinite article and the adjective other; the consequence of this for the modern language is that the existence of the determinative another blocks the co-occurrence of the indefinite article and other as separate syntactic constituents: *an other book. Determinatives other than the indefinite article precede other without such compounding [ . . . ]
You can combine most determiners with the adjective other freely:
the other book ← OK; the is a determinative in determiner function
my other book ← OK; my is a genitive pronoun in determiner function
But you can't combine an and other this way:
*an other book ← ungrammatical
another book ← OK; another is a determinative in determiner function
Is this just spelling? No:
the three books ← OK
the three other books ← OK
the other three books ← OK*a three books ← ungrammatical
*a three other books ← ungrammatical
*an other three books ← ungrammaticalanother three books ← OK
As you can see, another is not simply the combination of an other. It has its own patterns of usage unlike any other word in the English language and requires its own grammatical description.
Importantly, it's a single word in speech, too. We normally connect words in speech, and if an other were grammatical, it would usually be pronounced the same way as another:
an other /əˈnʌðər/ (connected speech)
another /əˈnʌðər/ (connected speech)
So in normal speech, we have no indication of whether it's one word or two.
But speakers can optionally pronounce words separately, if they like. For example, the other can be pronounced as two separate words with a pause in between, using the strong form of each word:
the other /ˈðiː ˈʌðər/ (pronounced separately, each word emphasized)
the other /ðiˈʌðər/ (pronounced together)
But this doesn't work with another. The strong form of an is not available, and the /n/ cannot be separated from the second syllable:
an other /ˈæn ˈʌðər/ ← not OK
another /əˈnʌðər/ ← OK
As you can see, this word is not only inseparable in writing, but in speech as well. This gives us a solid argument that another should be considered a single word, not just an arbitrary spelling of two words.
In fact, many native speakers are unaware that another used to be made of an + other. Because the /n/ is always pronounced as the onset of the second syllable, some speakers have even re-split the word as a nother, as in the colloquial American idiom a whole nother.
Similar things have happened in the past. A napron became an apron, and now everyone knows the word as apron. Few speakers today have ever heard the word napron. Would it be logical to say something like this?
apron = a napron - an ← true diachronically; not true synchronically
Sure, it was true historically, but it doesn't make any sense when you consider the modern language by itself. The same is true of the equation you found:
another = an + other ← true diachronically; not true synchronically
It's true that there are some combinations where the two words an other appear in sequence. This happens, for example, when other doesn't directly modify the head noun, but appears as part of a larger phrase:
an [other than honorable] discharge ← OK
Here, an is followed by a phrase which contains other. That phrase does include the word other, but that's okay. Cases like this are possible, but they're quite rare.
It's also okay if other is being used as a noun, although this doesn't happen very much:
The psychiatrist is emotionally and spiritually dead. He has no purpose or cause to give his life significance. One of T. S. Eliot’s “hollow men,” he lacks enthusiasm for his vocation, not believing that he has been called by an Other. (source)
Quotes like these are okay too. So how can we formulate a general rule that allows "exceptions" like these but disallows *an other? Something like this:
The sequence *an other is ungrammatical when an is in determiner function and other is an adjective in attributive function. The single word another is generally used instead.
