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A083696 a(n) = Sum_{r=0..2^(n-1)} (5^r/(2r)!)*Product_{k=0..2r-1} (2^n - k). 2
1, 6, 56, 6016, 72318976, 10460064284409856, 218825889667954898996994670329856, 95769539977943941232017762100658986141884645207653888255921750016 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
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
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 *)
PROG
(Sage) [2^(2^n -1)*lucas_number2(2^n, 1, -1) for n in (0..8)] # G. C. Greubel, Jan 14 2022
(Magma) [2^(2^n -1)*Lucas(2^n): n in [0..8]]; // G. C. Greubel, Jan 14 2022
CROSSREFS
Sequence in context: A268760 A137032 A053421 * A288680 A181430 A281557
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), May 22 2003
STATUS
approved

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Last modified March 5 09:07 EST 2024. Contains 370539 sequences. (Running on oeis4.)