For example, in triangular numbers $1, 3, 6, 10, 15, 21, 28, 36, 45, 55\dots$, two evens come after two odds. I want something like it for three odds and three evens.
closed as too broad by BaronVT, vadim123, Nicholas R. Peterson, Jesko Hüttenhain, Jack D'Aurizio Jan 10 at 18:36
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The sequence $a_n = \lceil n/3\rceil$ works.
The notation $\lceil m \rceil$ indicates the smallest integer not greater than $m$ (the "ceiling"). So the sequence for $n = 1, 2, 3, ...$ is $a_n = 1, 1, 1, 2, 2, 2, 3, 3, 3, ...$
Or, the more technical name for this sequence is the positive integers repeated three times. (Which is actually an incorrect name, as each integer is repeated just twice; it's written, it's repeated, and it's repeated a second time. But who's counting?)