A290565 - OEIS

%I #70 Sep 24 2021 03:11:16

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

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

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

%N Decimal expansion of sum of reciprocal golden rectangle numbers.

%C The constant k in A277266 such that A277266(n) ~ k*n.

%F Equals Sum_{n>=1} 1/(Fibonacci(n)*Fibonacci(n+1)).

%F Equals lim_{n->infinity} A277266(n)/n.

%F Equals 2 * (Sum_{k>=1} 1/(phi^k * F(k))) - 1/phi = 2 * A265290 - A094214, where phi is the golden ratio (A001622) and F(k) is the k-th Fibonacci number (A000045). - _Amiram Eldar_, Oct 05 2020

%F Equals 3/2 + 10*c*Integral_{x=0..infinity} f(x) dx, where c = sqrt(5)/log(phi) = A002163/A002390, phi = (1+sqrt(5))/2 = A001622, and f(x) = sin(x)/((exp(Pi*x/(2*log(phi)))-1)*(7-2*cos(x))*(3+2*cos(x))). - _Gleb Koloskov_, Sep 12 2021

%e 1/(1*1) + 1/(1*2) + 1/(2*3) + 1/(3*5) + ... = 1 + 1/2 + 1/6 + 1/15 + ... = 1.77387758328513234380...

%t RealDigits[ Sum[1/(Fibonacci[k]*Fibonacci[k + 1]), {k, 265}], 10, 111][[1]]

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

%Y Cf. A000045, A000796, A001622, A001654, A002163, A002390, A079586, A094214, A265290, A277266.

%K cons,nonn

%O 1,2

%A _Bobby Jacobs_ and _Robert G. Wilson v_, Aug 06 2017

%E More terms from _Alois P. Heinz_, Aug 06 2017