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A270582 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(k+1).

%I A270582 #9 Feb 16 2025 08:33:31
%S A270582 2,3,5,65,6529,136091233,41625259047416909,
%T A270582 2189507051227161558033650829868135,
%U A270582 75931290362065676573711484986356332365619562746656079489987281066955
%N A270582 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(k+1).
%C A270582 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 A270582 See A269993 for a guide to related sequences.
%H A270582 Clark Kimberling, <a href="/A270582/b270582.txt">Table of n, a(n) for n = 1..12</a>
%H A270582 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270582 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270582 sqrt(2) - 1 = 1/(2*2) + 1/(3*3) + 1/(4*5) + 1/(5*65) + ...
%t A270582 r[k_] := 1/(k+1); f[x_, 0] = x; z = 10;
%t A270582 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270582 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270582 x = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]
%Y A270582 Cf. A269993.
%K A270582 nonn,frac,easy
%O A270582 1,1
%A A270582 _Clark Kimberling_, Apr 03 2016