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A270584 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r(k) = 1/(k+1).

%I A270584 #9 Feb 16 2025 08:33:31
%S A270584 1,3,37,1204,21029921,425355555167420,439183524292095499600664584581,
%T A270584 240317442633783387248198509182959563857071128274317237128901,
%U A270584 1816763565571992723556609635427913847146292698536599340539742991592182627925499061514094793847919952134648005118828414904
%N A270584 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r(k) = 1/(k+1).
%C A270584 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 A270584 See A269993 for a guide to related sequences.
%H A270584 Clark Kimberling, <a href="/A270584/b270584.txt">Table of n, a(n) for n = 1..11</a>
%H A270584 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270584 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270584 tau - 1 = 1/(2*1) + 1/(3*3) + 1/(4*37) + 1/(5*1204) + ...
%t A270584 r[k_] := 1/(k+1); f[x_, 0] = x; z = 10;
%t A270584 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270584 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270584 x = GoldenRatio - 1; Table[n[x, k], {k, 1, z}]
%Y A270584 Cf. A269993.
%K A270584 nonn,frac,easy
%O A270584 1,2
%A A270584 _Clark Kimberling_, Apr 03 2016