A083697 a(n) = 2^(2^n - 1) * Fibonacci(2^n).

1, 2, 24, 2688, 32342016, 4677882957791232, 97861912906883207538212742365184, 42829440312913272520181533609472356498655100482256687829780267008

Offset

0,2

Comments

A083696(n)/a(n) converges to sqrt(5).

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10

Formula

a(n) = 2*a(n-1)*A083696(n-1).

a(n) = A058635(n) * A058891(n).

a(n) = 2^(2^n - 1) * A000045(2^n).

a(n) = Sum_{r=0..(2^n -1)} (5^r/(2*r+1)!)*Product_{k=0..2*r} (2^n - k).

Mathematica

Table[Sum[Product[2^n -k, {k, 0, 2*r}]k^r/(2*r+1)!, {r, 0, 2^n -1}], {n, 0, 8}]
Table[2^(2^n -1)*Fibonacci[2^n], {n, 0, 8}] (* G. C. Greubel, Jan 14 2022 *)

Prog

(Sage) [2^(2^n -1)*lucas_number1(2^n, 1, -1) for n in (0..8)] # G. C. Greubel, Jan 14 2022
(Magma) [2^(2^n -1)*Fibonacci(2^n): n in [0..8]]; // G. C. Greubel, Jan 14 2022

Crossrefs

Cf. A000045, A058635, A058891, A081460, A083696.

Sequence in context: A262206 A240993 A007079 * A110131 A112332 A101339

Adjacent sequences: A083694 A083695 A083696 * A083698 A083699 A083700

Keyword

easy,nonn

Author

Mario Catalani (mario.catalani(AT)unito.it), May 22 2003

Extensions

The next term is too large to include.

Better description from Ralf Stephan, Aug 29 2004

Status

approved