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A270349 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1,1/2,1/4,1/8,...)

%I A270349 #13 Feb 16 2025 08:33:31
%S A270349 3,7,27,650,689392,1130869248534,2046949388776880512222550,
%T A270349 5664769376602746621028306587399157369622446276283,
%U A270349 61600875764518391286867927949695082949269716944423018977948114995142883041085134431474743108010213
%N A270349 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1,1/2,1/4,1/8,...)
%C A270349 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 A270349 See A269993 for a guide to related sequences.
%H A270349 Clark Kimberling, <a href="/A270349/b270349.txt">Table of n, a(n) for n = 1..11</a>
%H A270349 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270349 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F A270349 a(n) = A270347(n+1).
%e A270349 sqrt(2) - 1 = 1/3 + 1/(2*7) + 1/(4*27) + ...
%t A270349 r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
%t A270349 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270349 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270349 x = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]
%o A270349 (PARI) r(k) = 2/2^k;
%o A270349 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270349 a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 18 2016
%Y A270349 Cf. A269993, A270347.
%K A270349 nonn,frac,easy
%O A270349 1,1
%A A270349 _Clark Kimberling_, Mar 17 2016