A005267 a(n) = -1 + a(0)*a(1)*...*a(n-1) with a(0) = 3.

3, 2, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, 106735757048926752040856495274871386126283608869, 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068029

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

The next term is too large to include.

An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004

Let u(k), v(k) be defined by u(1)=1, v(1)=3, u(k+1)=v(k)-u(k), v(k+1)=u(k)v(k); then a(n)=v(2n). - Benoit Cloitre, Apr 02 2002

For positive n, a(n) has digital root 2 or 5 depending on whether n is odd or even. (T. Koshy) - Lekraj Beedassy, Apr 11 2005

References

• R. K. Guy and R. Nowakowski, "Discovering primes with Euclid," Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
• T. Koshy, "Intriguing Properties Of Three Related Number Sequences", in Journal of Recreational Mathematics, Vol. 32(3) pp. 210-213, 2003-2004 Baywood NY.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..12
• R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
• Index entries for sequences of form a(n+1)=a(n)^2 + ...

Crossrefs

Cf. A000058, A000289, A117805.

Programs

PARI

a(n)=if(n<2,3*(n>=0)-(n>0),a(n-1)^2+a(n-1)-1)

Python

def a(n):
    if n == 0: return 2
    t = a(n-1)
    l = t+1
    u = t
    return l * u - 1
print([a(n) for n in range(0, 8)]) # Darío Clavijo, Aug 24 2024

Formula

a(n) = -1 + a(0)a(1)...a(n-1).

a(n) = -1 + Product_{iHenry Bottomley, Jul 31 2000

a(n+1) = a(n)^2 + a(n) - 1 if n>1. a(0)=3, a(1)=2.

An induction shows that a(n+1) = A117805(n) - 1. - R. J. Mathar, Apr 22 2007; M. F. Hasler, May 04 2007

For n>0, a(n) = a(0)^2 + a(1)^2 + ... + a(n-1)^2 - n - 6. - Max Alekseyev, Jun 19 2008