The alpha parameter controls the recurrence weights. Comparing two alpha choices shows how the arithmetic responds when the current input receives a smaller or larger share.
highlighted = computed this step
Alpha as weight
Alpha 1/2 gives the current observed value more weight than alpha 1/3. Motivation: alpha is the exact multiplier that decides how much the current input can move the state.
α=1/2α=1/3
Half prefix
With alpha 1/2, the first outputs are 2, 3, and 3. Why: each output follows the same recurrence with a fixed weight, so the sequence can be recomputed exactly.
ES1/2=2,3,3
Third prefix
With alpha 1/3, the first outputs are 2, 8/3, and 25/9. Interpretation: the smaller current-input weight leans more on accumulated history; this is a mechanical difference, not a tuning win by itself.
ES1/3=2,8/3,25/9
Diagram note
Honesty note: These values are exact formula outputs on the pinned integer series; they do not validate what happens later. Pixel positions are rounded for layout; every number shown is exact.