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"Fibonacci in digits - up and down": start with a(1)=1, a(2)=1; repeatedly adjoin either the sum of the two previous terms (if that sum happens to be odd) or else adjoin digits of the sum of previous two terms (if that sum happens to be even).
... a(8)=a(6)+a(7), a(9)=left digit of (a(7)+a(8)=13+21=3 4) as 34 is even, a(10)=right digit of (a(7)+a(8)=13+21=3 4) as 34 is even, a(13)=a(9)+a(10) as odd, ...