This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A255353 #37 Jan 05 2025 19:51:40
%S A255353 2,3,6,15,24,40,104,168,273,714,1155,1870,4895,7920,12816,33552,54288,
%T A255353 87841,229970,372099,602070,1576239,2550408,4126648,10803704,17480760,
%U A255353 28284465,74049690,119814915,193864606,507544127,821223648,1328767776
%N A255353 Denominators in an expansion of 3 - sqrt(5) as a sum of fractions +-1/d.
%C A255353 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.
%H A255353 Colin Barker, <a href="/A255353/b255353.txt">Table of n, a(n) for n = 1..1000</a>
%H A255353 Mohammad K. Azarian, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbAugust2013.pdf">The Value of a Series of Reciprocal Fibonacci Numbers, Problem B-1133</a>, Fibonacci Quarterly, Vol. 51, No. 3, August 2013, p. 275; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnAug14.pdf">Solution</a> published in Vol. 52, No. 3, August 2014, pp. 277-278.
%H A255353 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,8,0,0,-8,0,0,1).
%F A255353 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).
%F A255353 From _Colin Barker_, Dec 17 2015: (Start)
%F A255353 a(n) = 8*a(n-3) - 8*a(n-6) + a(n-9) for n>9.
%F A255353 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)).
%F A255353 (End)
%e A255353 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).
%t A255353 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 *)
%o A255353 (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
%Y A255353 Cf. A131561, A187799.
%K A255353 nonn,frac,easy
%O A255353 1,1
%A A255353 _Mohammad K. Azarian_, Feb 21 2015