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A270396 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/Fibonacci(k+1).

%I A270396 #15 Feb 16 2025 08:33:31
%S A270396 3,7,36,1040,1378784,2783678150237,20812561896916543523976387,
%T A270396 398006071848302987834283599453836703483929049938762,
%U A270396 105246367677020752496441044566935490666701848819994695873528056638957197400663802988967689301303582936
%N A270396 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/Fibonacci(k+1).
%C A270396 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 A270396 See A269993 for a guide to related sequences.
%H A270396 Clark Kimberling, <a href="/A270396/b270396.txt">Table of n, a(n) for n = 1..11</a>
%H A270396 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270396 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270396 sqrt(2) - 1 = 1/3 + 1/(2*7) + 1/(3*36) + 1/(5*1040) + ...
%t A270396 r[k_] := 1/Fibonacci[k+1]; f[x_, 0] = x; z = 10;
%t A270396 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270396 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270396 x = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]
%o A270396 (PARI) r(k) = 1/fibonacci(k+1);
%o A270396 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270396 a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 22 2016
%Y A270396 Cf. A269993, A000045, A188582.
%K A270396 nonn,frac,easy
%O A270396 1,1
%A A270396 _Clark Kimberling_, Mar 22 2016