cp's OEIS Frontend

A058181 Quadratic recurrence a(n) = a(n-1)^2 - a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.

%I A058181 #43 Jan 09 2025 09:37:34
%S A058181 1,0,-1,1,2,3,7,46,2109,4447835,19783236185116,
%T A058181 391376433956083065015485621,
%U A058181 153175513056180249189030531428945090978436751221570525
%N A058181 Quadratic recurrence a(n) = a(n-1)^2 - a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.
%H A058181 Vincenzo Librandi, <a href="/A058181/b058181.txt">Table of n, a(n) for n = 0..16</a>
%H A058181 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/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.
%H A058181 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%H A058181 <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1) = a(n)^2 + ...</a>
%F A058181 a(n)^2 = a(n+1) + a(n-1), a(-1-n) = a(n).
%F A058181 For n >= 4, a(n) = ceiling(c^(2^n)) with c=1.0303497388742578142745024606710866\
%F A058181 16436302563960998408889321488508667424048981473368773165340730475719244472111...
%F A058181 and c^(1/4) = 1.0075025785879710605024343257517358... - _Benoit Cloitre_, Apr 16 2007
%e A058181 a(6) = a(5)^2 - a(4) = 3^2 - 2 = 7.
%t A058181 Join[{a=1,b=0},Table[c=b^2-a;a=b;b=c,{n,13}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 18 2011 *)
%t A058181 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 A058181 (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 A058181 (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 A058181 (Sage)
%o A058181 def a(n):
%o A058181     if (n==0): return 1
%o A058181     elif (n==1): return 0
%o A058181     else: return a(n-1)^2 - a(n-2)
%o A058181 [a(n) for n in (0..15)] # _G. C. Greubel_, Jun 09 2019
%o A058181 (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 A058181 Cf. A058182.
%K A058181 sign
%O A058181 0,5
%A A058181 _Henry Bottomley_, Nov 15 2000