A333088 a(n) is the numerator of Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)).

1, 7, 143, 4351, 814001, 1304114687, 8811986820779, 5052800260335941, 153317149364862950801, 131408899191108437793754033, 11009306212815764937387730291387, 4837569887867603346019952058036959933, 37818210546715267110622871226615561517197713

Offset: 1

Author

A.H.M. Smeets, Mar 07 2020

Keywords

• nonn
• frac

Comments

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).

Examples

			These infinite sums begin: 1, 7/18, 143/960, ...
		

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..62
• Brother Alfred Brousseau, Summation of Infinite Fibonacci Series, The Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 143-168. See (5) and (6) p. 148.
• Stanley Rabinowitz, Algorithmic summation of reciprocals of products of Fibonacci numbers, The Fibonacci Quarterly, Vol. 37 (1999), pp. 122-127. See (23) and (25) p. 5.

Crossrefs

Cf. A105393, A290565, A333089 (denominator).

Programs

Mathematica

a[n_] := Numerator[Sum[1/(Fibonacci[2i-1]*Fibonacci[2i]),{i,1,n}]/Fibonacci[2n]]; Array[a, 13] (* Amiram Eldar, Mar 10 2020 *)

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)
    f0, f1 = 2*f0+f1, f0+f1 # A.H.M. Smeets, May 16 2020

Formula

a(n) = numerator of (1/Fibonacci(2n)) * Sum_{0 < i <= n} 1/(Fibonacci(2i-1)*Fibonacci(2i)).