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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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5
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I think the middle number is

$9$

West-to-East sequence

$7,9,20$
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

$1,9,10$
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

$1,9,19$
The smallest numbers which contain n Ns in their spelling (when written in English). The next number would be $99$.

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  • $\begingroup$ rot 13(BAR AVAR GRA pna or pbzcygrgryl qrfpevorq ol Ryrzragf B Ar A V Ar Gr A) Nyfb Svir, Avar naq Gjragl Fvk ner svefg guerr ahzoref pbagnvavat ibjryf V naq R $\endgroup$ – DEEM Feb 4 at 22:40
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    $\begingroup$ @DEEM rot13(Jung nobhg RVTUG (gung pbagnvaf V naq R)? V gnxr lbhe cbvag nobhg gur purzvpny flzobyf (jvyy rqvg nccebcevngryl).) $\endgroup$ – hexomino Feb 5 at 9:58
  • $\begingroup$ And 13,15,16,18,19,25 (I and E that is) $\endgroup$ – JMP Feb 5 at 15:29

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