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A270355 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1, 1/2, 1/4, 1/8, ...)

%I A270355 #16 Feb 16 2025 08:33:31
%S A270355 2,3,5,78,4962,15925310,303532967750376,72884922416996896007616951849,
%T A270355 3238110775186648021853203185875679911508503009261997468560,
%U A270355 7716186732679740909751872277405382774000613384297298421745471878603639986756704754013029661605882827711280194233739
%N A270355 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1, 1/2, 1/4, 1/8, ...)
%C A270355 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 A270355 See A269993 for a guide to related sequences.
%H A270355 Clark Kimberling, <a href="/A270355/b270355.txt">Table of n, a(n) for n = 1..12</a>
%H A270355 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270355 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270355 1/e = 1/2 + 1/(2*3) + 1/(4*5) + ...
%t A270355 r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
%t A270355 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270355 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270355 x = E - 2; Table[n[x, k], {k, 1, z}]
%o A270355 (PARI) r(k) = 2/2^k;
%o A270355 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270355 a(k, x=exp(1)-2) = ceil(r(k)/f(k-1, x));
%Y A270355 Cf. A269993.
%K A270355 nonn,frac,easy
%O A270355 1,1
%A A270355 _Clark Kimberling_, Mar 17 2016