A083696 - OEIS

%I #20 Jan 15 2022 09:47:17

%S 1,6,56,6016,72318976,10460064284409856,

%T 218825889667954898996994670329856,

%U 95769539977943941232017762100658986141884645207653888255921750016

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

%C 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.

%H G. C. Greubel, <a href="/A083696/b083696.txt">Table of n, a(n) for n = 0..10</a>

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

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

%F a(n) = 2^(2^n - 1) * Lucas(2^n). - _Vaclav Kotesovec_, Jan 08 2021

%t Table[Sum[Product[2^n - k, {k, 0, 2*r - 1}]5^r/(2*r)!, {r, 0, 2^(n - 1)}], {n, 0, 8}]

%t Table[2^(2^n - 1)*LucasL[2^n], {n, 0, 8}] (* _Vaclav Kotesovec_, Jan 08 2021 *)

%o (Sage) [2^(2^n -1)*lucas_number2(2^n, 1, -1) for n in (0..8)] # _G. C. Greubel_, Jan 14 2022

%o (Magma) [2^(2^n -1)*Lucas(2^n): n in [0..8]]; // _G. C. Greubel_, Jan 14 2022

%Y Cf. A000032, A083697, A081459.

%K easy,nonn

%O 0,2

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