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%I #8 Jan 15 2022 09:47:51
%S 1,2,24,2688,32342016,4677882957791232,
%T 97861912906883207538212742365184,
%U 42829440312913272520181533609472356498655100482256687829780267008
%N a(n) = 2^(2^n - 1) * Fibonacci(2^n).
%C A083696(n)/a(n) converges to sqrt(5).
%C Similar to A081460: a(n) is the denominator of the same mapping f(r)=(1/2)(r+5/r) but with initial value r=1.
%H G. C. Greubel, <a href="/A083697/b083697.txt">Table of n, a(n) for n = 0..10</a>
%F a(n) = 2*a(n-1)*A083696(n-1).
%F a(n) = A058635(n) * A058891(n).
%F a(n) = 2^(2^n - 1) * A000045(2^n).
%F a(n) = Sum_{r=0..(2^n -1)} (5^r/(2*r+1)!)*Product_{k=0..2*r} (2^n - k).
%t Table[Sum[Product[2^n -k, {k,0,2*r}]k^r/(2*r+1)!, {r,0,2^n -1}], {n,0,8}]
%t Table[2^(2^n -1)*Fibonacci[2^n], {n,0,8}] (* _G. C. Greubel_, Jan 14 2022 *)
%o (Sage) [2^(2^n -1)*lucas_number1(2^n, 1, -1) for n in (0..8)] # _G. C. Greubel_, Jan 14 2022
%o (Magma) [2^(2^n -1)*Fibonacci(2^n): n in [0..8]]; // _G. C. Greubel_, Jan 14 2022
%Y Cf. A000045, A058635, A058891, A081460, A083696.
%K easy,nonn
%O 0,2
%A Mario Catalani (mario.catalani(AT)unito.it), May 22 2003
%E The next term is too large to include.
%E Better description from _Ralf Stephan_, Aug 29 2004
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