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A045777 a(1)=1, a(2)=2; thereafter successive products of pairs of digits make further digits.

%I A045777 #19 Feb 15 2024 14:20:06
%S A045777 1,2,2,4,8,3,2,2,4,6,4,8,2,4,2,4,3,2,1,6,8,8,8,1,2,6,2,6,4,8,6,4,6,4,
%T A045777 8,2,1,2,1,2,1,2,2,4,3,2,4,8,2,4,2,4,2,4,3,2,1,6,2,2,2,2,2,2,4,8,1,2,
%U A045777 6,8,3,2,1,6,8,8,8,8,8,1,2,6,2,6,1,2,4,4,4,4,4,8,3,2,8,2,1,2,4,8,2,4,6,2,6
%N A045777 a(1)=1, a(2)=2; thereafter successive products of pairs of digits make further digits.
%C A045777 The numbers 0, 5, 7, and 9 never appear, but arbitrarily long sequences of 8's appear.
%H A045777 T. D. Noe, <a href="/A045777/b045777.txt">Table of n, a(n) for n = 1..15212</a>
%H A045777 Erich Friedman, <a href="https://erich-friedman.github.io/puzzle/mathpuzzle/">Puzzles of the Week</a>.
%e A045777 1*2=2 2*2=4 2*4=8 4*8=32 8*3=24...
%t A045777 t = {1, 2}; Do[ t = Join[t, IntegerDigits[t[[n-1]] t[[n-2]]]], {n, 3, 100}]; t
%o A045777 (Python)
%o A045777 from itertools import islice
%o A045777 from collections import deque
%o A045777 def agen(): # generator of terms
%o A045777     a = deque([1, 2])
%o A045777     while True:
%o A045777         a.extend(list(map(int, str(a[0]*a[1]))))
%o A045777         yield a.popleft()
%o A045777 print(list(islice(agen(), 105))) # _Michael S. Branicky_, Feb 15 2024
%K A045777 easy,nonn,base
%O A045777 1,2
%A A045777 _Erich Friedman_