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

1,2

Comments

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

Links

Table of n, a(n) for n=1..102.

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A063736 A212299 A193751 * A319263 A318386 A318334

Adjacent sequences: A290562 A290563 A290564 * A290566 A290567 A290568

Keyword

cons,nonn

Author

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

Extensions

More terms from Alois P. Heinz, Aug 06 2017

Status

approved