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%I M3789 N1544 #64 Dec 11 2023 19:02:38
%S 1,5,9,49,2209,4870849,23725150497409,562882766124611619513723649,
%T 316837008400094222150776738483768236006420971486980609
%N A nonlinear recurrence: a(0) = 1, a(1) = 5, a(n) = a(n-1)^2 - 4*a(n-1) + 4 for n>1.
%C An infinite coprime sequence defined by recursion. - _Michael Somos_, Mar 14 2004
%C 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
%C 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
%D Derek Jennings, Some reciprocal summation identities with applications to the Fibonacci and Lucas numbers, in: G. E. Bergum, Applications of Fibonacci Numbers, Vol. 7, Bergum G. E. et al. (eds.), Kluwer Academic Publishers, 1998, pp. 197-200.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000324/b000324.txt">Table of n, a(n) for n = 0..12</a>
%H A. V. Aho and N. J. A. Sloane, <a href="https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437, <a href="http://neilsloane.com/doc/doubly.html">alternative link</a>.
%H Daniel Duverney, <a href="https://www.ms.u-tokyo.ac.jp/journal/abstract/jms080206.html">Irrationality of Fast Converging Series of Rational Numbers</a>, Journal of Mathematical Sciences-University of Tokyo, Vol. 8, No. 2 (2001), pp. 275-316.
%H Daniel Duverney and Takeshi Kurosawa, <a href="https://doi.org/10.1007/s40993-022-00366-1">Transcendence of infinite products involving Fibonacci and Lucas numbers</a>, Research in Number Theory, Vol. 8 (2002), Article 68.
%H Solomon W. Golomb, <a href="http://www.jstor.org/stable/2311857">On certain nonlinear recurring sequences</a>, Amer. Math. Monthly, Vol. 70, No. 4 (1963), 403-405.
%H Romeo Meštrović, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
%H Seppo Mustonen, <a href="http://www.survo.fi/papers/resseq.pdf">On integer sequences with mutual k-residues</a>, 2005.
%H Seppo Mustonen, <a href="/A000215/a000215.pdf">On integer sequences with mutual k-residues</a>, 2005. [Local copy]
%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>.
%F a(n) = L(2^n)+2, if n>0 where L() is Lucas sequence.
%F For n>=1, a(n) = 4 + Product_{i=0..n-1} a(i). - _Vladimir Shevelev_, Dec 08 2010
%F From _Amiram Eldar_, Sep 10 2022: (Start)
%F a(n) = Lucas(2^(n-1))^2 for n > 1.
%F Sum_{n>=1} 4^n/a(n) = 4 (Jennings, 1998; Duverney, 2001). (End)
%F Product_{n>=1} (1 - 3/a(n)) = 1/4 (Duverney and Kurosawa, 2022). - _Amiram Eldar_, Jan 07 2023
%t t = {1, 5}; Do[AppendTo[t, t[[-1]]^2 - 4*t[[-1]] + 4], {n, 11}] (* _T. D. Noe_, Jun 19 2012 *)
%t 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 *)
%t Join[{1},NestList[#^2-4#+4&,5,10]] (* _Harvey P. Dale_, Dec 11 2023 *)
%o (PARI) a(n)=if(n<2,max(0,1+4*n),a(n-1)^2-4*a(n-1)+4)
%o (PARI) a(n)=if(n<1,n==0,n=2^n;fibonacci(n+1)+fibonacci(n-1)+2)
%Y a(n) = A001566(n-1)+2 (for n>0).
%Y Cf. A000032, A000058, A000215, A000289.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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