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A001696 a(n) = a(n-1)*(1 + a(n-1) - a(n-2)), a(0) = 0, a(1) = 1.

%I A001696 M1268 N0487 #48 Jan 09 2025 08:21:55
%S A001696 0,1,2,4,12,108,10476,108625644,11798392680793836,
%T A001696 139202068568601568785946949658348,
%U A001696 19377215893777651167043206536157529523359277782016064519251404524
%N A001696 a(n) = a(n-1)*(1 + a(n-1) - a(n-2)), a(0) = 0, a(1) = 1.
%C A001696 Also, numbers remaining after the following sieving process: In step 1, keep all numbers of the set N={0,1,2,...}. In step 2, keep only every second number after a(2)=2: N'={0,1,2,4,6,8,10,...}. In step 3, keep every 4th of the numbers following a(3)=4, N"={0,1,2,4,12,20,28,...}. In step 4, keep every 12th of the numbers beyond a(4)=12: {0,1,2,4,12,108,204,...}. In step 5, keep every 108th of the numbers beyond a(5)=108: {0,1,2,4,12,108,10476,...}, and so on. The next "gap" a(n+1)-a(n) is always a(n) times the former gap, i.e., a(n+1)-a(n) = a(n)*(a(n)-a(n-1)). - _M. F. Hasler_, Oct 28 2010
%C A001696 Number of plane trees where the root has fewer than n children and the i-th child of any node has fewer than i children. - _David Eppstein_, Dec 18 2021
%D A001696 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001696 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001696 John Cerkan, <a href="/A001696/b001696.txt">Table of n, a(n) for n = 0..13</a>
%H A001696 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
%H A001696 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%H A001696 <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%F A001696 a(n) ~ c^(2^n), where c = 1.15552822483840350150537253088299651035583896919522349372370013726451673646... . - _Vaclav Kotesovec_, May 21 2015
%t A001696 a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n-1]*(1 + a[n-1] - a[n-2]); Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Jul 02 2013 *)
%o A001696 (PARI) a(n)=if(n<2,n>0,a(n-1)*(1+a(n-1)-a(n-2)))
%o A001696 (Haskell)
%o A001696 a001696 n = a001696_list !! n
%o A001696 a001696_list = 0 : 1 : zipWith (-)
%o A001696    (zipWith (+) a001696_list' $ map (^ 2) a001696_list')
%o A001696    (zipWith (*) a001696_list a001696_list')
%o A001696    where a001696_list' = tail a001696_list
%o A001696 -- _Reinhard Zumkeller_, Apr 29 2013
%Y A001696 a(n)=A039941(2*n); first difference sequence of this sequence is A001697. - _Michael Somos_, May 19 2000
%K A001696 nonn,easy
%O A001696 0,3
%A A001696 _N. J. A. Sloane_