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%I #23 Jun 10 2016 00:21:00
%S 69,78,87,96,98,169,178,187,196,619,696,718,787,817,872,873,878,916,
%T 961,962,969,1169,1178,1691,1781,2987,6911,6916,6961,6962,6969,7817,
%U 7872,7873,7878,8117,8787,9116,9696,9878
%N Numbers without digit 0 or 5 whose "waterfall sequence" ends in 0,0,0,...
%C The "waterfall" sequence S for a given starting value S(1) is defined as S(n)=d(n-1)*d(n) (n>1), where d(n) is the n-th digit of the sequence.
%C When a(0) has a digit 0 or 5, then S is likely to end up in repeating zeros, which is the motivation for the definition of this sequence.
%H E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/Waterfalls.htm">Waterfalls (of multiplications)</a>, Mar 27 2012
%H E. Angelini, <a href="/A210614/a210614.pdf">Waterfalls (of multiplications)</a> [Cached copy, with permission]
%e The waterfall sequence for S(1)=69 is S=(69,54,45,20,16,20,10,0,0,6,12, 0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,2,0,0,0,...) with S(2)=6*9=54, S(3)=9*5=45, S(4)=5*4=20, etc.
%e The last "2" is obtained as 1*2 from the digits of term S(27)=12, thereafter there are no two consecutive nonzero digits and therefore only 0's can follow.
%e Similarly, for S(1)=78, one has S=(78,56,40,30,24,0,0,0,0,8,0,0,0,...), and only zeros thereafter since d(10)=4 is the last nonzero digit having a nonzero neighboring digit (d(9)=2, which yields S(10)=2*4=8).
%o (PARI) is_A210614(n)={!setintersect(["0","5"],Set(Vec(Str(n)))) & is_A210652(n)}
%o for(n=10,9999, is_A210614(n) & print1(n", "))
%K nonn,base
%O 1,1
%A _M. F. Hasler_, Mar 27 2012
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