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A270397 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/Fibonacci(k+1).

%I A270397 #21 Feb 16 2025 08:33:31
%S A270397 2,3,6,21,411,120274,10572781147,74407087111123560666,
%T A270397 5372512080606517833291366730287672914459,
%U A270397 41169436260792910821230360026041473906108740980452651576082359437785122898819171
%N A270397 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/Fibonacci(k+1).
%C A270397 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 A270397 See A269993 for a guide to related sequences.
%H A270397 Clark Kimberling, <a href="/A270397/b270397.txt">Table of n, a(n) for n = 1..13</a>
%H A270397 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>.
%H A270397 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270397 sqrt(3) - 1 = 1/2 + 1/(2*3) + 1/(3*6) + 1/(5*21) + ...
%t A270397 r[k_] := 1/Fibonacci[k+1]; f[x_, 0] = x; z = 10;
%t A270397 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270397 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270397 x = Sqrt[3] - 1; Table[n[x, k], {k, 1, z}]
%o A270397 (PARI) r(k) = 1/fibonacci(k+1);
%o A270397 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270397 a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 22 2016
%Y A270397 Cf. A269993, A000045, A160390.
%K A270397 nonn,frac,easy
%O A270397 1,1
%A A270397 _Clark Kimberling_, Mar 22 2016