%I #52 Jun 28 2024 14:54:50

%S 1,2,3,7,43,1807,3263443,10650056950807,113423713055421844361000443,

%T 12864938683278671740537145998360961546653259485195807

%N A variant of Sylvester's sequence: a(0)=1 and for n>0, a(n) = (a(0)*a(1)*...*a(n-1)) + 1.

%C A variant of A000058, starting with an extra 1.

%D Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, 1997, Exercice 3.3.4 page 284.

%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/10.4169/college.math.j.42.4.329">Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958</a>, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.

%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/10.4169/college.math.j.43.4.337">Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution</a> College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.

%H Vjekoslav Kovač, <a href="https://arxiv.org/abs/2406.17593">On simultaneous rationality of two Ahmes series</a>, arXiv:2406.17593 [math.NT], 2024.

%F For n>0, a(n) = A000058(n-1).

%F a(1) = 2, a(n+1) = a(n)^2 - a(n) + 1. a(n) = round(c^(2^n)), where c = 1.264... is the Vardi constant, A076393. - _Thomas Ordowski_, Jun 11 2013

%F From _Bernard Schott_, Apr 06 2021: (Start)

%F Sum_{n>=0} 1/a(n) = 2.

%F Sum_{n>=0} (-1)^(n+1)/a(n) = 2 * (A118227 - 1). (End)

%t a[0] = 1; a[n_] := a[n] = Product[a[k], {k, 0, n - 1}] + 1

%o (Haskell)

%o a129871 n = a129871_list !! n

%o a129871_list = 1 : a000058_list -- _Reinhard Zumkeller_, Dec 18 2013

%Y Cf. A000058 which is the main entry for this sequence.

%Y Cf. A118227.

%K nonn

%O 0,2

%A _Ben Branman_, Sep 16 2011

%E Corrected and rewritten by _Ben Branman_, Sep 16 2011

%E Edited by _Max Alekseyev_, Oct 11 2012