A083696 a(n) = Sum_{r=0..2^(n-1)} (5^r/(2r)!)*Product_{k=0..2r-1} (2^n - k).

1, 6, 56, 6016, 72318976, 10460064284409856, 218825889667954898996994670329856, 95769539977943941232017762100658986141884645207653888255921750016

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Similar to A081459: a(n) is the numerator 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

Crossrefs

Cf. A000032, A083697, A081459.

Programs

Magma

[2^(2^n -1)*Lucas(2^n): n in [0..8]]; // G. C. Greubel, Jan 14 2022

Mathematica

Table[Sum[Product[2^n - k, {k, 0, 2*r - 1}]5^r/(2*r)!, {r, 0, 2^(n - 1)}], {n, 0, 8}]
Table[2^(2^n - 1)*LucasL[2^n], {n, 0, 8}] (* Vaclav Kotesovec, Jan 08 2021 *)

Sage

[2^(2^n -1)*lucas_number2(2^n, 1, -1) for n in (0..8)] # G. C. Greubel, Jan 14 2022

Formula

a(n)/A083697(n) converges to sqrt(5).

a(n) = a(n-1)^2 + 5*A083697(n-1)^2.

a(n) = 2^(2^n - 1) * Lucas(2^n). - Vaclav Kotesovec, Jan 08 2021