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A269995 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1,1/2,1/3,1/4,...)

%I A269995 #13 Feb 16 2025 08:33:30
%S A269995 3,7,36,1300,2206054,14887222782418,292542996759533035472424790,
%T A269995 7282957087563143077864043818232331102110274520711753058,
%U A269995 259880230781524461939787525796521055875618560291171401151227648777033604862236784108033156713828890456025177451
%N A269995 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1,1/2,1/3,1/4,...)
%C A269995 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 A269995 See A269993 for a guide to related sequences.
%H A269995 Clark Kimberling, <a href="/A269995/b269995.txt">Table of n, a(n) for n = 1..12</a>
%H A269995 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A269995 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A269995 sqrt(2) - 1 = 1/(2*3) + 1/(3*7) + 1/(4*36) + ...
%t A269995 r[k_] := 1/k; f[x_, 0] = x; z = 10;
%t A269995 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A269995 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A269995 x = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]
%Y A269995 Cf. A269993.
%K A269995 nonn,frac,easy
%O A269995 1,1
%A A269995 _Clark Kimberling_, Mar 15 2016