Count the Cows

Farmer John's cows have spread themselves out in his largest pasture, which can be represented as a large 2D grid of square cells (imagine a huge chessboard).

The pattern of cows across the pasture is quite intriguing. For every cell $(x,y)$ with x \ge 0 and y \ge 0, there is a cow at $(x,y)$ if, for all integers k \ge 0, the parity of the remainders when \left\lfloor \frac{x}{3^k} \right\rfloor and \left\lfloor \frac{y}{3^k} \right\rfloor are divided by $3$ is the same. In other words: both remainders are odd (equal to $1$), or both are even (equal to $0$ or $2$). For example, the cells among 0 \le x, y < 9 that contain cows are marked with ones in the diagram below:

        x
    012345678

  0 101000101
  1 010000010
  2 101000101
  3 000101000
y 4 000010000
  5 000101000
  6 101000101
  7 010000010
  8 101000101

Farmer John is curious how many cows are present in certain diagonal regions of his pasture. He asks $q$ queries, each consisting of three integers x_i, y_i, d_i. For each query, Farmer John wants to know how many cows are located at the cells along the diagonal from (x_i, y_i) to (x_i + d_i, y_i + d_i) (endpoints inclusive).