A255353 Denominators in an expansion of 3 - sqrt(5) as a sum of fractions +-1/d.

2, 3, 6, 15, 24, 40, 104, 168, 273, 714, 1155, 1870, 4895, 7920, 12816, 33552, 54288, 87841, 229970, 372099, 602070, 1576239, 2550408, 4126648, 10803704, 17480760, 28284465, 74049690, 119814915, 193864606, 507544127, 821223648, 1328767776

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

Cf. A131561, A187799.

Sequence in context: A066653 A081945 A329745 * A248652 A158027 A346776

Adjacent sequences: A255350 A255351 A255352 * A255354 A255355 A255356

Keyword

nonn,frac,easy

Author

Mohammad K. Azarian, Feb 21 2015

Status

approved