Build a staircase for my kid

Python 2, 55 bytes

i=2
exec"print(i%2*' '+`2%i*1122`*i)[:i];i+=1;"*input()

Try it online!

Cycles between blocks of 22, 44, except the top row is 00. For example, on input 10, prints

00
 22
2244
 2244
224422
 224422
22442244
 22442244
2244224422
 2244224422

Prints rows of increasing length i=2,3,.. by creating a space for odd lengths, repeating the pattern i times, and trunctating to length i. The pattern is 2244 for all rows except the first one i=2 for which it's 0. This is achieved with the arithmetic expression 2%i*1122.

Jelly,  21 19  16 bytes

d2SR+%3x2⁶;ṙḂµ€Y

A full program printing the result.

Uses 00, 11, and 22 as the blocks.

Try it online!

How?

d2SR+%3x2⁶;ṙḂµ€Y - Main link: number n
             µ€  - for €ach "row" in n (i.e. for row, r = 1 to n inclusive):
d2               -   divmod 2   -> [integer-divide by 2, remainder] i.e. [r/2, r%2]
  S              -   sum        -> r/2 + r%2
   R             -   range      -> [1, 2, 3, ..., r/2 + r%2]
    +            -   add r      -> [r+1, r+2, r+3, ..., r + r/2 + r%2]
     %3          -   modulo 3   -> [r%3+1, r%3+2, r%3+0, ..., (r + r/2 + r%2)%3]
                 -   e.g.: r: 1  , 2  , 3    , 4    , 5      , 6      , 7       , ...
                             [2], [0], [1,2], [2,0], [0,1,2], [1,2,0], [2,0,1,2], ...
       x2        -   times 2 - repeat each twice (e.g. [2,0,1,2] -> [2,2,0,0,1,1,2,2]
         ⁶       -   literal space character
          ;      -   concatenate (add a space character to the left)
            Ḃ    -   r mod 2 (1 if the row is odd, 0 if it is even (1 at the top))
           ṙ     -   rotate (the list) left by (r mod 2)
               Y - join with newlines
                 - implicit print (no brackets printed due to the presence of characters)

JavaScript (ES6), 80 bytes

n=>eval(`for(s=11,i=1;i++<n;)s+='\\n'+(' '+'2233'.repeat(n)).substr(i%2,i+1);s`)

f=
n=>eval(`for(s=11,i=1;i++<n;)s+='\\n'+(' '+'2233'.repeat(n)).substr(i%2,i+1);s`)
console.log(f(+prompt()))

JavaScript (ES6), 87 bytes

Previous solution.

n=>[11,...Array(n).fill(' '+'2233'.repeat(n)).map((r,n)=>r.slice(n%2,n+3+n%2))].join`
`