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A000324
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A nonlinear recurrence: a(n) = a(n-1)^2 - 4*a(n-1) + 4 (for n>1).
(Formerly M3789 N1544)
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8
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OFFSET
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0,2
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COMMENTS
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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, A000215, 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
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REFERENCES
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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).
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LINKS
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T. D. Noe, 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, alternative link.
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
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ...
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FORMULA
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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
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MATHEMATICA
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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-François Alcover, Feb 07 2016 *)
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PROG
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(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)
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CROSSREFS
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a(n) = A001566(n-1)+2 (for n>0).
Cf. A000058.
Sequence in context: A328333 A173776 A289909 * A123817 A124421 A262918
Adjacent sequences: A000321 A000322 A000323 * A000325 A000326 A000327
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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