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A270352 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/2, 1/4, 1/8, ...)

%I A270352 #16 Feb 16 2025 08:33:31
%S A270352 4,8,44,977,498723,138012074956,45087947486104434546449,
%T A270352 2223745971024423874814212532278502253766982404,
%U A270352 3439676840537267257806008796995789895364959784333600339427716437786254731225969490712842205
%N A270352 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/2, 1/4, 1/8, ...)
%C A270352 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 A270352 See A269993 for a guide to related sequences.
%H A270352 Clark Kimberling, <a href="/A270352/b270352.txt">Table of n, a(n) for n = 1..11</a>
%H A270352 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270352 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270352 1/Pi = 1/4 + 1/(2*8) + 1/(4*44) + ...
%t A270352 r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
%t A270352 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270352 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270352 x = 1/Pi; Table[n[x, k], {k, 1, z}]
%o A270352 (PARI) r(k) = 2/2^k;
%o A270352 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270352 a(k, x=1/Pi) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 18 2016
%Y A270352 Cf. A269993.
%K A270352 nonn,frac,easy
%O A270352 1,1
%A A270352 _Clark Kimberling_, Mar 17 2016