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A270520 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/k!.

%I A270520 #10 Feb 16 2025 08:33:31
%S A270520 2,3,3,5,6,14,28,352,18574,44518346,400826881311158,
%T A270520 25342673472297507115832358714,
%U A270520 62130292590921086469117151395751018383242940308998211770
%N A270520 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/k!.
%C A270520 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 A270520 See A269993 for a guide to related sequences.
%H A270520 Clark Kimberling, <a href="/A270520/b270520.txt">Table of n, a(n) for n = 1..20</a>
%H A270520 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270520 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270520 sqrt(3) - 1 = 1/(1*2) + 1/(2*3) + 1/(6*3) + 1/(24*5) + ...
%t A270520 r[k_] := 1/k!; f[x_, 0] = x; z = 10;
%t A270520 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270520 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270520 x = Sqrt[3] - 1; Table[n[x, k], {k, 1, z}]
%o A270520 (PARI) r(k) = 1/k!;
%o A270520 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270520 a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 31 2016
%Y A270520 Cf. A269993, A000142, A160390.
%K A270520 nonn,frac,easy
%O A270520 1,1
%A A270520 _Clark Kimberling_, Mar 30 2016