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Denominators of r-Egyptian fraction expansion for 1/e, where r(k) = 1/(2k-1).
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%I #11 Feb 16 2025 08:33:31

%S 3,10,165,218673,75510967206,14666670996451472494064,

%T 318033435047744040119174255756277946082958110,

%U 222562499295932133989982996162129528076446080094832884826693648678455802606574139206041317

%N Denominators of r-Egyptian fraction expansion for 1/e, where r(k) = 1/(2k-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="/A270553/b270553.txt">Table of n, a(n) for n = 1..11</a>

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

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

%e 1/e = 1/(1*3) + 1/(3*10) + 1/(5*165) + 1/(7*218673) + ...

%t r[k_] := 1/(2k-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/(2*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)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Apr 03 2016

%Y Cf. A269993, A005408, A068985.

%K nonn,frac,easy,changed

%O 1,1

%A _Clark Kimberling_, Apr 02 2016