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A270548 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(2k-1).

%I A270548 #11 Feb 16 2025 08:33:31
%S A270548 3,5,15,163,29203,1370794960,5693192315226228214,
%T A270548 247405800822801380465687897681838336769,
%U A270548 267682228701778523205506744045084667800917057557706608910309126004853790212423
%N A270548 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(2k-1).
%C A270548 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 A270548 See A269993 for a guide to related sequences.
%H A270548 Clark Kimberling, <a href="/A270548/b270548.txt">Table of n, a(n) for n = 1..12</a>
%H A270548 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270548 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270548 sqrt(2) - 1 = 1/(1*3) + 1/(3*5) + 1/(5*15) + 1/(7*163) + ...
%t A270548 r[k_] := 1/(2k-1); f[x_, 0] = x; z = 10;
%t A270548 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270548 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270548 x = Sqrt(2) - 1; Table[n[x, k], {k, 1, z}]
%o A270548 (PARI) r(k) = 1/(2*k-1);
%o A270548 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270548 a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Apr 03 2016
%Y A270548 Cf. A269993, A005408, A188582.
%K A270548 nonn,frac,easy
%O A270548 1,1
%A A270548 _Clark Kimberling_, Apr 02 2016