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A000324 A nonlinear recurrence: a(n) = a(n-1)^2 - 4*a(n-1) + 4 (for n>1).
(Formerly M3789 N1544)
8
1, 5, 9, 49, 2209, 4870849, 23725150497409, 562882766124611619513723649, 316837008400094222150776738483768236006420971486980609 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

This is the special case k=4 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

A000058, A00215, A000289 and this sequence here can be represented as values of polynomials defined via P_0(z)= 1+z, P_{n+1}(z) = z+ prod_{i=0..n} P_i(z), with recurrences P_{n+1}(z) = (P_n(z))^2 -z*P_n(z) +z, n>=0. - Vladimir Shevelev, Dec 08 2010

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

T. D. Noe, Table of n, a(n) for n = 0..12

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.

S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.

R. Mestrovic, 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

S. Mustonen, On integer sequences with mutual k-residues

Index entries for sequences of form a(n+1)=a(n)^2 + ...

FORMULA

a(n) = L(2^n)+2, if n>0 where L() is Lucas sequence.

For n>=1, a(n) = 4+Prod{i=0,...,n-1} a(i). - Vladimir Shevelev, Dec 08 2010

MATHEMATICA

t = {1, 5}; Do[AppendTo[t, t[[-1]]^2 - 4*t[[-1]] + 4], {n, 11}] (* T. D. Noe, Jun 19 2012 *)

Join[{1}, RecurrenceTable[{a[n] == a[n-1]^2 - 4*a[n-1] + 4, a[1] == 5}, a, {n, 1, 8}]] (* Jean-Fran├žois Alcover, Feb 07 2016 *)

PROG

(PARI) a(n)=if(n<2, max(0, 1+4*n), a(n-1)^2-4*a(n-1)+4)

(PARI) a(n)=if(n<1, n==0, n=2^n; fibonacci(n+1)+fibonacci(n-1)+2)

CROSSREFS

a(n) = A001566(n-1)+2 (for n>0).

Cf. A000058.

Sequence in context: A100457 A080872 A173776 * A123817 A124421 A262918

Adjacent sequences:  A000321 A000322 A000323 * A000325 A000326 A000327

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 30 08:42 EDT 2017. Contains 284300 sequences.