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%I #56 Jan 05 2023 03:54:38
%S 0,1,1,2,3,7,16,65,321,4546,107587,20773703,11595736272,
%T 431558332068481,134461531248108526465,186242594112190847520182173826,
%U 18079903385772308300945867582153787570051,34686303861638264961101080464895364211215702792496667048327
%N a(n) = a(n-1) + a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.
%H T. D. Noe, <a href="/A000278/b000278.txt">Table of n, a(n) for n = 0..25</a>
%H W. Duke, Stephen J. Greenfield, and Eugene R. Speer, <a href="https://cs.uwaterloo.ca/journals/JIS/green2/qf.html">Properties of a Quadratic Fibonacci Recurrence</a>, J. Integer Seq. 1 (1998), Article #98.1.8.
%F a(2n) is asymptotic to A^(sqrt(2)^(2n-1)) where A=1.668751581493687393311628852632911281060730869124873165099170786836201970866312366402366761987... and a(2n+1) to B^(sqrt(2)^(2n)) where B=1.693060557587684004961387955790151505861127759176717820241560622552858106116817244440438308887... See reference for proof. - _Benoit Cloitre_, May 03 2003
%p A000278 := proc(n) option remember; if n <= 1 then n else A000278(n-2)^2+A000278(n-1); fi; end;
%t Join[{a=0,b=1},Table[c=a^2+b;a=b;b=c,{n,16}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 22 2011 *)
%t RecurrenceTable[{a[n +2] == a[n +1] + a[n]^2, a[0] == 1, a[1] == 1}, a, {n, 0, 16}] (* _Robert G. Wilson v_, Apr 14 2017 *)
%o (PARI) a(n)=if(n<2,n>0,a(n-1)+a(n-2)^2)
%o (Sage)
%o def A000278():
%o x, y = 0, 1
%o while True:
%o yield x
%o x, y = x + y, x * x
%o a = A000278(); [next(a) for i in range(18)] # _Peter Luschny_, Dec 17 2015
%o (Magma) [n le 2 select n-1 else Self(n-1) + Self(n-2)^2: n in [1..18]]; // _Vincenzo Librandi_, Dec 17 2015
%Y Cf. A000283, A058182.
%K nonn
%O 0,4
%A Stephen J. Greenfield (greenfie(AT)math.rutgers.edu)
%E Name edited by _Petros Hadjicostas_, Nov 03 2019
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