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A129871

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

1, 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807

OFFSET

0,2

COMMENTS

A variant of A000058, starting with an extra 1.

REFERENCES

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

LINKS

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.

Vjekoslav Kovač, On simultaneous rationality of two Ahmes series, arXiv:2406.17593 [math.NT], 2024.

FORMULA

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

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

From Bernard Schott, Apr 06 2021: (Start)

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

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

MATHEMATICA

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

PROG

(Haskell)

a129871 n = a129871_list !! n

a129871_list = 1 : a000058_list -- Reinhard Zumkeller, Dec 18 2013

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Ben Branman, Sep 16 2011

EXTENSIONS

Corrected and rewritten by Ben Branman, Sep 16 2011

Edited by Max Alekseyev, Oct 11 2012

STATUS

approved

URL: https://oeis.org/A129871