A324007 - OEIS

%I #18 Jan 05 2025 19:51:41

%S 7,1,0,8,5,5,3,5,1,4,2,9,3,2,8,4,1,6,8,8,7,6,9,4,4,9,0,3,8,4,2,7,0,8,

%T 3,3,0,4,5,1,1,8,0,4,8,4,1,0,3,0,8,6,3,9,9,7,4,9,7,3,5,1,4,9,3,6,9,6,

%U 4,2,3,8,2,6,1,1,3,5,4,4,8,4,1,7,5,8,8,4,1,6,8,1,7,1,4,8,5,8,5,7,6,8,5,4,9

%N Decimal expansion of the sum of reciprocals of the products of 3 consecutive Fibonacci numbers.

%H Brother Alfred Brousseau, <a href="https://www.mathstat.dal.ca/FQ/Scanned/7-2/brousseau1.pdf">Summation of Infinite Fibonacci Series</a>, The Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 143-168.

%H R. S. Melham, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/41-1/melham.pdf">On Some Reciprocal Sums of Brousseau: An Alternative Approach to That of Carlitz</a>, Fibonacci Quarterly, Vol. 41, No. 1 (2003), pp. 59-62.

%F From _Amiram Eldar_, Feb 09 2023: (Start)

%F Equals Sum_{k>=1} 1/A065563(k).

%F Equals 1 - A158933 (Melham, 2003). (End)

%e 0.71085535142932841688769449038427083304511804841030863997497351493696423826...

%t RealDigits[ Sum[ N[ 1/Product[ Fibonacci@j, {j, k, k + 2}], 128], {k, 177}], 10, 111][[1]]

%o (PARI) suminf(n=1, 1/(fibonacci(n)*fibonacci(n+1)*fibonacci(n+2))) \\ _Michel Marcus_, Feb 19 2019

%Y Cf. A000045, A065563, A079586, A158933, A290565, A322711, A324008.

%K nonn,cons

%O 0,1

%A _Robert G. Wilson v_, Feb 11 2019