A270394 - OEIS

%I #15 Feb 24 2018 10:11:07

%S 2,3,9,59,9437,62059971,2813586350787717,

%T 8534689167911295735140758101600,

%U 54171527001975050997893888972139886506909953999125751170768531

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

%C 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 See A269993 for a guide to related sequences.

%H Clark Kimberling, <a href="/A270394/b270394.txt">Table of n, a(n) for n = 1..12</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%e sqrt(1/2) = 1/2 + 1/(2*3) + 1/(3*9) + 1/(5*59) + ...

%t r[k_] := 1/Fibonacci[k+1]; f[x_, 0] = x; z = 10;

%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

%t x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]

%o (PARI) r(k) = 1/fibonacci(k+1);

%o f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););

%o a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 22 2016

%Y Cf. A269993, A000045, A010503.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Mar 22 2016