R, (削除) 69 (削除ここまで) 58 bytes

function(n)3-n%in%(l=cumsum(1:n))-n%in%outer(c(0,l),l,"+")

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Explanation:

function(n){
 T <- cumsum(1:n) # first n triangular numbers [1,3,6]
 S <- outer(c(0,T),T,"+") # sums of the first n triangular numbers,
 # AND the first n triangular numbers [1,3,6,2,4,7,4,6,9,7,9,12]
 3 - n %in% S - n %in% T # if n is in T, it's also in S, so it's 3-2: return 1
 # if n is in S but not T, it's 3-1: return 2
 # if n isn't in S, it's not in T, so 3-0: return 3
}

R, (削除) 69 (削除ここまで) 58 bytes

function(n)3-n%in%(l=cumsum(1:n))-n%in%outer(c(0,l),l,"+")

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Explanation:

function(n){
 T <- cumsum(1:n) # first n triangular numbers [1,3,6]
 S <- outer(c(0,T),T,"+") # sums of the first n triangular numbers,
 # AND the first n triangular numbers [1,3,6,2,4,7,4,6,9,7,9,12]
 3 - (n %in% S) - (n %in% T) # if n is in T, it's also in S, so it's 3-2: return 1
 # if n is in S but not T, it's 3-1: return 2
 # if n isn't in S, it's not in T, so 3-0: return 3
}

R, 69 bytes

function(n)"if"(n%in%(l=cumsum(1:n)),1,"if"(n%in%outer(l,l,"+"),2,3))

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cumsum(1:n) is the first n triangular numbers. If n is in l, then we return 1. outer(l,l,"+") is all sums of pairs of triangular numbers. If n is in that, then it's the sum of 2 triangular numbers. Otherwise, n must be the sum of at least 3 triangular numbers, so we return 3.

R, (削除) 69 (削除ここまで) 58 bytes

function(n)3-n%in%(l=cumsum(1:n))-n%in%outer(c(0,l),l,"+")

Try it online!

Explanation:

function(n){
 T <- cumsum(1:n) # first n triangular numbers [1,3,6]
 S <- outer(c(0,T),T,"+") # sums of the first n triangular numbers,
 # AND the first n triangular numbers [1,3,6,2,4,7,4,6,9,7,9,12]
 3 - n %in% S - n %in% T # if n is in T, it's also in S, so it's 3-2: return 1
 # if n is in S but not T, it's 3-1: return 2
 # if n isn't in S, it's not in T, so 3-0: return 3
}