A003096 a(n) = a(n-1)^2 - 1, a(0) = 2.

2, 3, 8, 63, 3968, 15745023, 247905749270528, 61457260521381894004129398783, 3776994870793005510047522464634252677140721938309041881088

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

After a(0) = 2 and a(1) = 3, this can never be prime, since a(n) = (a(n-1) + 1) * (a(n-1) - 1). Each term is relatively prime to its successor. - Jonathan Vos Post, Jun 06 2008

Mensa (see Dutch link below) indicates high intelligence by offering a self test containing a number of problems, one of which is "Complete each series with the element that logically continues the series: 3968, 63, 8, 3". - David A. Corneth, May 19 2024

References

• R. K. Guy, How to factor a number, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 49-89.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..12
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
• Mensa, self-test indicative for high IQ
• Index entries for sequences of form a(n+1)=a(n)^2 + ...

Crossrefs

Cf. A003095, A077124, A139244.

Programs

Magma

[n le 1 select 2 else Self(n-1)^2 -1: n in [1..12]]; // G. C. Greubel, Oct 27 2022

Maple

a := proc(n) local k, v: v := 2: for k from 1 to n do v := v^2-1: od: v: end:
seq(a(n), n = 0 .. 8); # Lorenzo Sauras Altuzarra, Feb 01 2023

Mathematica

NestList[#^2-1&,2,10] (* Harvey P. Dale, Nov 06 2011 *)

PARI

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

SageMath

def A003096(n): return 2 if (n==0) else A003096(n-1)^2 -1
[A003096(n) for n in range(12)] # G. C. Greubel, Oct 27 2022

Formula

a(n-1) = ceiling(c^(2^n)) where c = 1.295553... = A077124. - Benoit Cloitre, Nov 29 2002