OFFSET
1,1
COMMENTS
The minus sign in front of a fraction is considered the sign of the numerator and hence the sign of the fraction does not appear in this sequence. We note that numerators are in A131561.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Mohammad K. Azarian, The Value of a Series of Reciprocal Fibonacci Numbers, Problem B-1133, Fibonacci Quarterly, Vol. 51, No. 3, August 2013, p. 275; Solution published in Vol. 52, No. 3, August 2014, pp. 277-278.
Index entries for linear recurrences with constant coefficients, signature (0,0,8,0,0,-8,0,0,1).
FORMULA
3 - sqrt(5) = Sum_{n>=1} 1/(F(2*n)*F(2*n+1)) + 1/(F(2*n)*F(2*n+2)) - 1/(F(2*n+1)*F(2*n+2)), where F = A000045 (Fibonacci numbers).
From Colin Barker, Dec 17 2015: (Start)
a(n) = 8*a(n-3) - 8*a(n-6) + a(n-9) for n>9.
G.f.: x*(2+3*x+6*x^2-x^3-8*x^5+x^8) / ((1-x)*(1+x+x^2)*(1-7*x^3+x^6)).
(End)
EXAMPLE
1/(1*2) + 1/(1*3) - 1/(2*3) + 1/(3*5) + 1/(3*8) - 1/(5*8) + 1/(8*13) + 1/(8*21) - 1/(13*21) + 1/(21*34) + 1/(21*55) - 1/(34*55) + ... + = 3 - sqrt(5).
MATHEMATICA
Table[SeriesCoefficient[x (2 + 3 x + 6 x^2 - x^3 - 8 x^5 + x^8)/((1 - x) (1 + x + x^2) (1 - 7 x^3 + x^6)), {x, 0, n}], {n, 33}] (* Michael De Vlieger, Dec 17 2015 *)
PROG
(PARI) Vec(x*(2+3*x+6*x^2-x^3-8*x^5+x^8)/((1-x)*(1+x+x^2)*(1-7*x^3+x^6)) + O(x^40)) \\ Colin Barker, Dec 17 2015
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Mohammad K. Azarian, Feb 21 2015
STATUS
approved