Chasing Need

Is it possible to have a row of 19 LEDs that are constantly lit but dimmed, that perform a chasing pattern up and down the row by individually illuminating to their maximum brightness? If so, what would be the necessary components?

This can be done with a WS2812B led strip.
See:

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Just an Uno and current limit resistors.
Can do PWM in software for fade up & down.