A270402 - OEIS

%I #13 Feb 24 2018 11:53:20

%S 2,3,7,51,1677,10015186,84570728548669,15258402443210459328886903125,

%T 491252531873843788731747079099102301138243342168362480520,

%U 7253884884686449148422066285941951777383387553675422580747925068364662702275919955251519389652837758517396514428571

%N Denominators of r-Egyptian fraction expansion for e - 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="/A270402/b270402.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 e - 2 = 1/2 + 1/(2*3) + 1/(3*7) + ...

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

%Y Cf. A269993, A000045, A001113.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Mar 22 2016