There are Four patterns below.
Two diagonal, one horizontal and one vertical. The middle number is common to them all
What is it? It is not mathematical per se.
NO partial answers. Explain all four patterns in your answer.
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I think the middle number is
These are numbers for which (in English) the letter N is in the nth last position and each number is the smallest such number which is greater than its predecessor. The next number would be $90$.
North-West to South-East sequence
These are the numbers which can be written in English solely using letters with value $1$ in Scrabble (E,A,I,O,N,R,T,L,S,U). The next number would be $19$.
As DEEM has pointed out, these are also the first three numbers whose English names can be broken into valid chemical symbols (e.g ONE = O Ne). I actually haven't found another number to fit into this sequence yet.
North to South sequence
$5, 9, 26$
The numbers which contain the letter I (in English) and have a square number of letters. The next number would be $31$.
North-East to South-West sequence
The smallest numbers which contain n Ns in their spelling (when written in English). The next number would be $99$.