A290565 Decimal expansion of sum of reciprocal golden rectangle numbers.

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, 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, 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

Offset: 1

Author

Bobby Jacobs and Robert G. Wilson v, Aug 06 2017

Keywords

• cons
• nonn

Comments

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

Examples

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

Crossrefs

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

Programs

Mathematica

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

PARI

suminf(n=1, 1/(fibonacci(n)*fibonacci(n+1))) \\ Michel Marcus, Feb 19 2019

Formula

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

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

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

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

Extensions

More terms from Alois P. Heinz, Aug 06 2017