This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A058182 #50 Jan 16 2024 02:46:37
%S A058182 1,0,1,1,2,5,27,734,538783,290287121823,84266613096281243382112,
%T A058182 7100862082718357559748563880517486086728702367,
%U A058182 50422242317787290639189291009890702507917377925161079229314384058371278254659634544914784801
%N A058182 a(n) = a(n-1)^2 + a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.
%C A058182 Has property that CONTINUANT([1, 1, 2, 5, 27, 734, 538783, ...]) = [1, 2, 5, 27, 734, 538783, ...]. - _N. J. A. Sloane_ Jul 19 2002
%C A058182 For n > 2, a(n) is the numerator of the simplified continued fraction resulting from [a(2), a(3), ..., a(n)]. Therefore, for n > 2, a(n) represents the number of ways to tile a (n-2)-board with dominoes and stackable squares, where nothing can be stacked on a domino but otherwise for 2 < i < n, the i-th cell may be stacked by as many as a(i) squares (see Benjamin, A. and Quinn, J.). - _Melvin Peralta_, Feb 22 2016
%D A058182 Arthur Benjamin and Jennifer Quinn, Proofs that Really Count, Mathematical Association of America, 2003, see pages 49-51.
%H A058182 Melvin Peralta, <a href="/A058182/b058182.txt">Table of n, a(n) for n = 0..15</a>
%H A058182 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A058182 <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%F A058182 a(n)^2 = a(n+1) - a(n-1), a(-1-n) = -a(n).
%F A058182 For n > 1, a(n+1) = floor(c^(2^n)) where c=1.108604586393628626769904017539.... - _Benoit Cloitre_, Nov 30 2002
%F A058182 a(n+1) = a(n)^2 + floor(sqrt(a(n))) = A000290(a(n)) + A000196(a(n)) for n > 2. - _Reinhard Zumkeller_, May 16 2006
%e A058182 a(6) = a(5)^2 + a(4) = 5^2 + 2 = 27.
%t A058182 Join[{a=1,b=0},Table[c=a+b^2;a=b;b=c,{n,12}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 22 2011 *)
%t A058182 Join[{1},Transpose[NestList[{Last[#],Last[#]^2+First[#]}&,{0,1},12]][[1]]] (* _Harvey P. Dale_, May 15 2011 *)
%t A058182 RecurrenceTable[{a[0] == 1, a[1] == 0, a[n] == a[n-1]^2 + a[n-2]}, a, {n, 13}] (* _Vincenzo Librandi_, Feb 23 2016 *)
%o A058182 (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 A058182 (Magma) I:=[1,0]; [n le 2 select I[n] else Self(n-1)^2+Self(n-2): n in [1..13]]; // _Vincenzo Librandi_, Feb 23 2016
%Y A058182 Cf. A000278, A005605, A058181.
%K A058182 nonn,nice,eigen
%O A058182 0,5
%A A058182 _Henry Bottomley_, Nov 15 2000
%E A058182 More terms from _Reinhard Zumkeller_, May 16 2006