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 A255353 Denominators in an expansion of 3 - sqrt(5) as a sum of fractions +-1/d. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 27 03:13 EDT 2022. Contains 354093 sequences. (Running on oeis4.)