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A001696 a(n) = a(n-1)*(1 + a(n-1) - a(n-2)), a(0) = 0, a(1) = 1.
(Formerly M1268 N0487)
4

%I M1268 N0487 #38 Dec 18 2021 14:57:38

%S 0,1,2,4,12,108,10476,108625644,11798392680793836,

%T 139202068568601568785946949658348,

%U 19377215893777651167043206536157529523359277782016064519251404524

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

%C 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)). [From _M. F. Hasler_, Oct 28 2010]

%C Number of plane trees where the root has fewer than n children and the ith child of any node has fewer than i children. - _David Eppstein_, Dec 18 2021

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H John Cerkan, <a href="/A001696/b001696.txt">Table of n, a(n) for n = 0..13</a>

%H A. V. Aho and N. J. A. Sloane, <a href="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, <a href="http://neilsloane.com/doc/doubly.html">alternative link</a>.

%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>

%F a(n) ~ c^(2^n), where c =

%F 1.15552822483840350150537253088299651035583896919522349372370013726451673646... . - _Vaclav Kotesovec_, May 21 2015

%t 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 (PARI) a(n)=if(n<2,n>0,a(n-1)*(1+a(n-1)-a(n-2)))

%o (Haskell)

%o a001696 n = a001696_list !! n

%o a001696_list = 0 : 1 : zipWith (-)

%o (zipWith (+) a001696_list' $ map (^ 2) a001696_list')

%o (zipWith (*) a001696_list a001696_list')

%o where a001696_list' = tail a001696_list

%o -- _Reinhard Zumkeller_, Apr 29 2013

%Y a(n)=A039941(2*n); first difference sequence of this sequence is A001697. - _Michael Somos_, May 19 2000

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)