Counting: 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Label of field: 1 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2I started wondering about these pairs of numbers. Obviously they are periodic. In general, the period of "Counting" is n, the period of "Label of field" is 2n-2, where n is the number of fields. The period of the pairs is LCM(2n-2,n). That lead me to:
Mathematica> Table[LCM[2*n-2,n], {n, 16}]
{0, 2, 12, 12, 40, 30, 84, 56, 144, 90, 220, 132, 312, 182, 420, 240}
This is EIS sequence
A045895.
Of course, all elements of this sequence are a multiple of 2. Dividing by 2 yields:
Mathematica> Table[LCM[2*n-2,n]/2, {n, 16}]
{0, 1, 6, 6, 20, 15, 42, 28, 72, 45, 110, 66, 156, 91, 210, 120}
This is EIS sequence
A045896.