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A270549 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/(2k-1).

%I A270549 #13 Feb 16 2025 08:33:31
%S A270549 2,2,4,10,102,9988,462079550,1246459580018549814,
%T A270549 3451767175159069042246539740614797183,
%U A270549 16047263805335625632784779620610026996218698731392917143951229224582015756
%N A270549 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/(2k-1).
%C A270549 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 A270549 See A269993 for a guide to related sequences.
%H A270549 Clark Kimberling, <a href="/A270549/b270549.txt">Table of n, a(n) for n = 1..13</a>
%H A270549 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270549 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270549 sqrt(3) - 1 = 1/(1*2) + 1/(3*2) + 1/(5*4) + 1/(7*10) + ...
%t A270549 r[k_] := 1/(2k-1); f[x_, 0] = x; z = 10;
%t A270549 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270549 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270549 x = Sqrt(3) - 1; Table[n[x, k], {k, 1, z}]
%o A270549 (PARI) r(k) = 1/(2*k-1);
%o A270549 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270549 a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Apr 03 2016
%Y A270549 Cf. A269993, A005408, A160390.
%K A270549 nonn,frac,easy
%O A270549 1,1
%A A270549 _Clark Kimberling_, Apr 02 2016