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A093094 "Products into digits": start with a(1)=2, a(2)=2; adjoin digits of product of a(k) and a(k+1) for k from 1 to infinity.

%I A093094 #23 Aug 18 2025 11:36:40
%S A093094 2,2,4,8,3,2,2,4,6,4,8,2,4,2,4,3,2,1,6,8,8,8,1,2,6,2,6,4,8,6,4,6,4,8,
%T A093094 2,1,2,1,2,1,2,2,4,3,2,4,8,2,4,2,4,2,4,3,2,1,6,2,2,2,2,2,2,4,8,1,2,6,
%U A093094 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
%N A093094 "Products into digits": start with a(1)=2, a(2)=2; adjoin digits of product of a(k) and a(k+1) for k from 1 to infinity.
%C A093094 Only the digits 1,2,3,4,6,8 occur, infinitely often. The sequence is not periodic. Around a(800) there are many 8's.
%C A093094 From _Giovanni Resta_, Mar 16 2006: (Start)
%C A093094 Proof that sequence is not periodic:
%C A093094 Let us assume that somewhere in the sequence there is a subsequence of 3 adjacent 8': ...,8,8,8,....(which is true).
%C A093094 Then we know that in the following there will be the subsequence ...,6,4,6,4.. (i.e. 8x8, 8x8) again, there will be somewhere ...,2,4,2,4,2,4,... (i.e. 6x4, 4x6, 6x4) and finally ...,8,8,8,8,8,...
%C A093094 Analogously, starting from 8,8,8,8 we obtain 6,4,6,4,6,4 then 2,4,2,4,2,4,2,4,2,4 and finally 8,8,8,8,8,8,8,8,8.
%C A093094 Generalizing, if somewhere appears a run of k>2 8's, then in some future position will appear a run of at least 4*k-7 8's (where since k>2, 4*k-7>k).
%C A093094 So the sequence will contain arbitrary long runs of 8's, without being constantly equal to 8, thus it cannot be periodic. (End)
%C A093094 Essentially the same as A045777. [_R. J. Mathar_, Sep 08 2008]
%H A093094 Reinhard Zumkeller, <a href="/A093094/b093094.txt">Table of n, a(n) for n = 1..10000</a>
%e A093094 a(3)=a(1)*a(2), a(4)=a(2)*a(3), a(5)=first digit of (a(3)*a(4)), a(6)=2nd digit of (a(3)*a(4)), a(9)=a(6)*a(7)
%t A093094 Fold[Join[#, IntegerDigits[Times @@ #[[#2;; #2+1]]]] &, {2, 2}, Range[100]] (* _Paolo Xausa_, Aug 18 2025 *)
%o A093094 (Haskell)
%o A093094 a093094 n = a093094_list !! (n-1)
%o A093094 a093094_list = f [2,2] where
%o A093094    f (u : vs@(v : _)) = u : f (vs ++
%o A093094      if w < 10 then [w] else uncurry ((. return) . (:)) $ divMod w 10)
%o A093094         where w = u * v
%o A093094 -- _Reinhard Zumkeller_, Aug 08 2013
%o A093094 (Python)
%o A093094 from itertools import islice
%o A093094 from collections import deque
%o A093094 def agen(): # generator of terms
%o A093094     a = deque([2, 2])
%o A093094     while True:
%o A093094         a.extend(list(map(int, str(a[0]*a[1]))))
%o A093094         yield a.popleft()
%o A093094 print(list(islice(agen(), 101))) # _Michael S. Branicky_, Feb 15 2024
%Y A093094 Cf. A045777, A093086, A093087, A093088, A093089, A093090, A093091, A093095, A093096, A093097, A096381.
%K A093094 nonn,base
%O A093094 1,1
%A A093094 _Bodo Zinser_, Mar 20 2004
%E A093094 Definition revised by _Franklin T. Adams-Watters_, Mar 16 2006