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A269996 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1,1/2,1/3,1/4,...)

%I A269996 #13 Feb 16 2025 08:33:30
%S A269996 2,3,6,26,939,800567,626897816036,732632470241183632257841,
%T A269996 31706715561023122142248280773186018287458544854469,
%U A269996 1666726692230759969765850044548001173784581299264219742879080654883940143766478552206863259848365362
%N A269996 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1,1/2,1/3,1/4,...)
%C A269996 Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1).  Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k).  Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.
%C A269996 See A269993 for a guide to related sequences.
%H A269996 Clark Kimberling, <a href="/A269996/b269996.txt">Table of n, a(n) for n = 1..13</a>
%H A269996 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A269996 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A269996 sqrt(3) - 1 = 1/2 + 1/(2*3) + 1/(3*6) + ...
%t A269996 r[k_] := 1/k; f[x_, 0] = x; z = 10;
%t A269996 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A269996 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A269996 x = Sqrt[3] - 1; Table[n[x, k], {k, 1, z}]
%Y A269996 Cf. A269993.
%K A269996 nonn,frac,easy
%O A269996 1,1
%A A269996 _Clark Kimberling_, Mar 15 2016