A000289 A nonlinear recurrence: a(n) = a(n-1)^2 - 3*a(n-1) + 3 (for n>1).

1, 4, 7, 31, 871, 756031, 571580604871, 326704387862983487112031, 106735757048926752040856495274871386126283608871, 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068031

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

This is the special case k=3 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

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..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!)
• S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
• Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
• Seppo Mustonen, On integer sequences with mutual k-residues
• Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
• Index entries for sequences of form a(n+1)=a(n)^2 + ...

Crossrefs

Cf. A000058.

Programs

Mathematica

Join[{1}, RecurrenceTable[{a[n] == a[n-1]^2 - 3*a[n-1] + 3, a[1] == 4}, a, {n, 1, 9}]] (* Jean-François Alcover, Feb 06 2016 *)

PARI

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

Formula

a(n) = A005267(n) + 2 (for n>0).

a(n) = ceiling(c^(2^n)) + 1 where c = A077141. - Benoit Cloitre, Nov 29 2002

For n>0, a(n) = 3 + Product_{i=0..n-1} a(i). - Vladimir Shevelev, Dec 08 2010