$$a_n=\{Y_n,X_n\}$$

where $X_n$ is sequence A319572 and $Y_n$ is sequence A319573 in the OEIS database. These are the coordinates of the stripe enumeration of $N \times N$ where $N = \{0, 1, 2, \ldots\}$.

$$a_n=\{Y_n,X_n\}$$

where $X_n$ is sequence A319572 and $Y_n$ is sequence A319573 in the OEIS database. These are the coordinates of the stripe enumeration of $N \times N$ where $N = \{0, 1, 2, \ldots\}$. A "Stripe Enumeration" function to produce these sequences is provided here.

Here is some Mathematica code to test for this:

f[{a_, b_}] := (1/2)*(1 - (-1)^(a + b))*{Max[0, a - 1], 
b + 1} + (1/2)*(1 + (-1)^(a + b))*{a + 1, Max[0, b - 1]}

RecurrenceTable[{a[n + 1] == f[a[n]], a[0] == {0, 0}}, a, {n, 0, 30}]

The output for $a_n$ is

_{{{0, 0}, {1, 0}, {0, 1}, {0, 2}, {1, 1}, {2, 0}, {3, 0}, {2, 1}, {1, 2}, {0, 3}, {0, 4}, {1, 3}, {2, 2}, {3, 1}, {4, 0}, {5, 0}, {4, 1}, {3, 2}, {2, 3}, {1, 4}, {0, 5}, {0, 6}, {1, 5}, {2, 4}, {3, 3}, {4, 2}, {5, 1}, {6, 0}, {7, 0}, {6, 1}, {5, 2}}}

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