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A333088
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a(n) is the numerator of Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)).
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4
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1, 7, 143, 4351, 814001, 1304114687, 8811986820779, 5052800260335941, 153317149364862950801, 131408899191108437793754033, 11009306212815764937387730291387, 4837569887867603346019952058036959933, 37818210546715267110622871226615561517197713
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OFFSET
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1,2
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COMMENTS
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The denominators are given in A333089.
Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)) is a fraction for n > 0.
Sum_{i > 0} 1/Fibonacci(i)^2, i.e., the n = 0 case, is known to be transcendental. See A105393.
Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n-1)) is an irrational number for n > 0. See for instance A290565 (n = 1).
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LINKS
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FORMULA
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a(n) = numerator of (1/Fibonacci(2n)) * Sum_{0 < i <= n} 1/(Fibonacci(2i-1)*Fibonacci(2i)).
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EXAMPLE
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These infinite sums begin: 1, 7/18, 143/960, ...
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MATHEMATICA
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a[n_] := Numerator[Sum[1/(Fibonacci[2i-1]*Fibonacci[2i]), {i, 1, n}]/Fibonacci[2n]]; Array[a, 13] (* Amiram Eldar, Mar 10 2020 *)
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PROG
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(PARI) a(n) = numerator(sum(i=1, n, 1/(fibonacci(2*i-1)*fibonacci(2*i)))/ fibonacci(2*n)); \\ Michel Marcus, Mar 10 2020
(Python)
from math import gcd
f0, f1, snum, sden, n = 1, 1, 0, 1, 0
while n < 13:
snum, sden, n = f0*f1*snum+sden, sden*f0*f1, n+1
d = gcd(snum, sden*f0)
print(n, snum//d)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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