%I #40 Sep 08 2022 08:45:02
%S 1,0,-1,1,2,3,7,46,2109,4447835,19783236185116,
%T 391376433956083065015485621,
%U 153175513056180249189030531428945090978436751221570525
%N Quadratic recurrence a(n) = a(n-1)^2 - a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.
%H Vincenzo Librandi, <a href="/A058181/b058181.txt">Table of n, a(n) for n = 0..16</a>
%H A. V. Aho and N. J. A. Sloane, <a href="http://neilsloane.com/doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart. 11 (1973), 429-437.
%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>, Fib. Quart. 11 (1973), 429-437. [See <a href="https://www.fq.math.ca/Scanned/11-4/aho-b.pdf">here</a> for the missing page.]
%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1) = a(n)^2 + ...</a>
%F a(n)^2 = a(n+1) + a(n-1), a(-1-n) = a(n).
%F For n >= 4, a(n) = ceiling(c^(2^n)) with c=1.0303497388742578142745024606710866\
%F 16436302563960998408889321488508667424048981473368773165340730475719244472111...
%F and c^(1/4) = 1.0075025785879710605024343257517358... - _Benoit Cloitre_, Apr 16 2007
%e a(6) = a(5)^2 - a(4) = 3^2 - 2 = 7.
%t Join[{a=1,b=0},Table[c=b^2-a;a=b;b=c,{n,13}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 18 2011 *)
%t RecurrenceTable[{a[0]==1, a[1]==0, a[n]==a[n-1]^2 - a[n-2]}, a, {n, 13}] (* _Vincenzo Librandi_, Nov 11 2012 *)
%o (PARI) a(n)=if(n<0, a(-1-n), if(n<2, 1-n, a(n-1)^2-a(n-2))) /* _Michael Somos_, May 05 2005 */
%o (Magma) I:=[1,0]; [n le 2 select I[n] else Self(n-1)^2 - Self(n-2): n in [1..15]]; // _G. C. Greubel_, Jun 09 2019
%o (Sage)
%o def a(n):
%o if (n==0): return 1
%o elif (n==1): return 0
%o else: return a(n-1)^2 - a(n-2)
%o [a(n) for n in (0..15)] # _G. C. Greubel_, Jun 09 2019
%o (GAP) a:=[1,0];; for n in [3..15] do a[n]:=a[n-1]^2-a[n-2]; od; a; # _G. C. Greubel_, Jun 09 2019
%Y Cf. A058182.
%K sign
%O 0,5
%A _Henry Bottomley_, Nov 15 2000