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A270519 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/k!.

%I A270519 #10 Feb 16 2025 08:33:31
%S A270519 3,7,18,217,21586,132830816,8232750479147118,
%T A270519 8738244742575919521189548340591,
%U A270519 28575128242342620144630216663972970082807062570299713849045286
%N A270519 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/k!.
%C A270519 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 A270519 See A269993 for a guide to related sequences.
%H A270519 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270519 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270519 sqrt(2) - 1 = 1/(1*3) + 1/(2*7) + 1/(6*18) + 1/(24*217) + ...
%t A270519 r[k_] := 1/k!; f[x_, 0] = x; z = 10;
%t A270519 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270519 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270519 x = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]
%o A270519 (PARI) r(k) = 1/k!;
%o A270519 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270519 a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 31 2016
%Y A270519 Cf. A269993, A000142, A188582.
%K A270519 nonn,frac,easy
%O A270519 1,1
%A A270519 _Clark Kimberling_, Mar 30 2016