Revision 7cc030be-8be7-460f-b72c-09c4c05e4aba - Code Golf Stack Exchange

##JavaScript (ES6), 52 bytes

<!-- language: lang-js -->

 f=(n,t=0)=>t<n?f(n-++t,t):t.toString(36)+(n?f(n):'')

###How?

Rather than explicitly computing ***T(i) = 1 + 2 + 3 + ... + i***, we start with ***t = 0*** and iteratively subtract ***t+1*** from ***n*** while ***t < n***, incrementing ***t*** at each iteration. When the condition is fulfilled, ***T(t)*** has been subtracted from ***n*** and the output is updated accordingly. We repeat the process until ***n = 0***.

Below is a summary of all operations for ***n = 100***.

 n | t | t < n | output
 ----+----+-------+--------
 100 | 0 | yes | ""
 99 | 1 | yes | ""
 97 | 2 | yes | ""
 94 | 3 | yes | ""
 90 | 4 | yes | ""
 85 | 5 | yes | ""
 79 | 6 | yes | ""
 72 | 7 | yes | ""
 64 | 8 | yes | ""
 55 | 9 | yes | ""
 45 | 10 | yes | ""
 34 | 11 | yes | ""
 22 | 12 | yes | ""
 9 | 13 | no | "d"
 ----+----+-------+--------
 9 | 0 | yes | "d"
 8 | 1 | yes | "d"
 6 | 2 | yes | "d"
 3 | 3 | no | "d3"
 ----+----+-------+--------
 3 | 0 | yes | "d3"
 2 | 1 | yes | "d3"
 0 | 2 | no | "d32"

###Test cases

<!-- begin snippet: js hide: true console: true babel: false -->

<!-- language: lang-js -->

 f=(n,t=0)=>t<n?f(n-++t,t):t.toString(36)+(n?f(n):'')

 console.log(f(1)) // 1
 console.log(f(2)) // 11
 console.log(f(3)) // 2
 console.log(f(4)) // 21
 console.log(f(5)) // 211
 console.log(f(6)) // 3
 console.log(f(100)) // d32
 console.log(f(230)) // k5211
 console.log(f(435)) // t
 console.log(f(665)) // z731

<!-- end snippet -->