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A270350 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1, 1/2, 1/4, 1/8, ...)

%I A270350 #16 Feb 16 2025 08:33:31
%S A270350 2,3,4,44,1446,3423518,263631451737996,70985515555913904515293113895,
%T A270350 8645798497265822420998718966216306501746531100894289290802,
%U A270350 78713180847550502513757221862401308079612732875925186430170968601702893264445327722349352410275677392885249561650440
%N A270350 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1, 1/2, 1/4, 1/8, ...)
%C A270350 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 A270350 See A269993 for a guide to related sequences.
%H A270350 Clark Kimberling, <a href="/A270350/b270350.txt">Table of n, a(n) for n = 1..12</a>
%H A270350 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270350 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270350 sqrt(3) - 1 = 1/2 + 1/(2*3) + 1/(4*4) + ...
%t A270350 r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
%t A270350 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270350 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270350 x = Sqrt[3] - 1; Table[n[x, k], {k, 1, z}]
%o A270350 (PARI) r(k) = 2/2^k;
%o A270350 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270350 a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 18 2016
%Y A270350 Cf. A269993.
%K A270350 nonn,frac,easy
%O A270350 1,1
%A A270350 _Clark Kimberling_, Mar 17 2016