A333208 a(n) is the numerator of Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)), with Lucas(i) as defined in A000032.

1, 7, 551, 48091, 2148268, 87644575267, 161577754532123, 36595152483523582367, 16965509829762630129638831, 173107561150078104051618631740949, 1108595900580419409151086339986148307, 2608169750203411467722731179728125652086612772

Offset

1,2

Comments

The denominators are given in A333209.

See A333088 and A333089 for similar fractions for infinite sums of Fibonacci numbers.

Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)) is a fraction for n > 0.

Sum_{i >= 0} 1/Lucas(i)^2 = 1/4 + A105394, i.e., the n = 0 case, is believed to be transcendental.

Links

Table of n, a(n) for n=1..12.

Formula

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

Lim_{n -> inf} (a(n)/A333209(n)) / (a(n-1)/A333209(n-1)) = 1 - 1/phi = 1/phi^2 = A132318.

The following generalization holds: (Start)

Let H_(a,b) (n) be defined by H_(a,b) (0) = a, H_(a,b) (1) = b and H_(a,b) (n) = H_(a,b) (n-1) + H_(a,b) (n-2) for n > 1, then

Sum_{i >= 0} 1/(H_(a,b) (i)*H_(a,b) (i+2n)) = (1/Fibonacci(2n)) * Sum_{i=1..n} 1/(H_(a,b) (2i-2)*H_(a,b) (2i-1)) for n > 0, and are thus all fractions. Specially, H_(0,1) are the Fibonacci numbers A000045, H_(2,1) as here, are the Lucas numbers A000032, and H_(3,1) are the Pibonacci numbers A104449. (End)

Example

These infinite sums begin: 1/2, 7/36, 551/7392, ...

Mathematica

a[n_] := Numerator[Sum[1/(LucasL[2 i - 2]*LucasL[2 i - 1]), {i, 1, n}]/Fibonacci[2 n]]; Array[a, 12] (* Amiram Eldar, Mar 11 2020 *)

Crossrefs

Cf. A000032, A105394, A132318, A333088, A333089, A333209.

Sequence in context: A126196 A182433 A329368 * A093169 A159029 A308582

Adjacent sequences: A333205 A333206 A333207 * A333209 A333210 A333211

Keyword

nonn

Author

A.H.M. Smeets, Mar 11 2020

Status

approved