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A101361 a(1) = a(2) = 1; for n > 2, a(n) = Knuth's Fibonacci (or circle) product "a(n-1) o a(n-2)".

%I A101361 #21 Aug 31 2024 08:33:29
%S A101361 1,1,3,8,55,987,121393,267914296,72723460248141,
%T A101361 43566776258854844738105,7084593923980518516849609894969925639,
%U A101361 690168906931029935139391829792095612517948949963798093315456
%N A101361 a(1) = a(2) = 1; for n > 2, a(n) = Knuth's Fibonacci (or circle) product "a(n-1) o a(n-2)".
%H A101361 Alois P. Heinz, <a href="/A101361/b101361.txt">Table of n, a(n) for n = 1..17</a>
%H A101361 D. E. Knuth, <a href="http://dx.doi.org/10.1016/0893-9659(88)90176-0">Fibonacci multiplication</a>, Appl. Math. Lett. 1 (1988), 57-60.
%F A101361 a(n) = Fibonacci(2*Fibonacci(n)).
%F A101361 Third-order nonlinear recursion: a(0)=1, a(1)=1, a(2)=3, a(n)=(a(n-1)^2 - a(n-2)^2)/a(n-3). - _T. D. Noe_, Mar 17 2009
%e A101361 1o1 = 3, 1o3 = 8, 3o8 = 55, ...
%p A101361 with(combinat); f:=n->fibonacci(2*fibonacci(n));
%p A101361 # second Maple program:
%p A101361 F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
%p A101361 a:= n-> F(2*F(n)):
%p A101361 seq(a(n), n=1..12);  # _Alois P. Heinz_, Jan 20 2017
%t A101361 Table[ Fibonacci[2Fibonacci[n]], {n, 12}] (* _Robert G. Wilson v_, Feb 12 2005 *)
%o A101361 (PARI) a(n)=if(n<1,0,fibonacci(2*fibonacci(n)))
%Y A101361 Cf. A000045, A101330.
%K A101361 nonn,easy
%O A101361 1,3
%A A101361 _N. J. A. Sloane_, Jan 26 2005
%E A101361 Formula and more terms from _Michael Somos_, Feb 03 2005