Ways to get to the number
Javascript ES6, 45 44 bytes
f=(a,b=B)=>a<(B=b)?f(a+1)+f(a+2)+f(a*3):a==b
Example runs:
f(1,2) -> 1
f(2,6) -> 6
f(1 10) -> 84
f=(a,b=B)=>a<(B=b)?f(a+1)+f(a+2)+f(a*3):a==b
Example runs:
f(1,2) -> 1
f(2,6) -> 6
f(1 10) -> 84