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A058182
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a(n) = a(n-1)^2 + a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.
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8
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1, 0, 1, 1, 2, 5, 27, 734, 538783, 290287121823, 84266613096281243382112, 7100862082718357559748563880517486086728702367, 50422242317787290639189291009890702507917377925161079229314384058371278254659634544914784801
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OFFSET
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0,5
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COMMENTS
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Has property that CONTINUANT([1, 1, 2, 5, 27, 734, 538783, ...]) = [1, 2, 5, 27, 734, 538783, ...]. - N. J. A. Sloane Jul 19 2002
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
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REFERENCES
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Arthur Benjamin and Jennifer Quinn, Proofs that Really Count, Mathematical Association of America, 2003, see pages 49-51.
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LINKS
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FORMULA
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a(n)^2 = a(n+1) - a(n-1), a(-1-n) = -a(n).
For n > 1, a(n+1) = floor(c^(2^n)) where c=1.108604586393628626769904017539.... - Benoit Cloitre, Nov 30 2002
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EXAMPLE
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a(6) = a(5)^2 + a(4) = 5^2 + 2 = 27.
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MATHEMATICA
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Join[{1}, Transpose[NestList[{Last[#], Last[#]^2+First[#]}&, {0, 1}, 12]][[1]]] (* Harvey P. Dale, May 15 2011 *)
RecurrenceTable[{a[0] == 1, a[1] == 0, a[n] == a[n-1]^2 + a[n-2]}, a, {n, 13}] (* Vincenzo Librandi, Feb 23 2016 *)
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PROG
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(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 */
(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
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CROSSREFS
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KEYWORD
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nonn,nice,eigen
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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