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

0, 1, 2, 4, 12, 108, 10476, 108625644, 11798392680793836, 139202068568601568785946949658348, 19377215893777651167043206536157529523359277782016064519251404524

Offset

0,3

Comments

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]

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

References

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

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

Links

John Cerkan, Table of n, a(n) for n = 0..13

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437, alternative link.

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Formula

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

1.15552822483840350150537253088299651035583896919522349372370013726451673646... . - Vaclav Kotesovec, May 21 2015

Mathematica

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 *)

Prog

(PARI) a(n)=if(n<2, n>0, a(n-1)*(1+a(n-1)-a(n-2)))
(Haskell)
a001696 n = a001696_list !! n
a001696_list = 0 : 1 : zipWith (-)
   (zipWith (+) a001696_list' $ map (^ 2) a001696_list')
   (zipWith (*) a001696_list a001696_list')
   where a001696_list' = tail a001696_list
-- Reinhard Zumkeller, Apr 29 2013

Crossrefs

a(n)=A039941(2*n); first difference sequence of this sequence is A001697. - Michael Somos, May 19 2000

Sequence in context: A038791 A327563 A326950 * A276534 A326969 A304986

Adjacent sequences: A001693 A001694 A001695 * A001697 A001698 A001699

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved