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A270371 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/4,1/9,1/16,...).

%I A270371 #15 Feb 16 2025 08:33:31
%S A270371 2,2,2,3,7,7702,1234163819,1590823281229385753,
%T A270371 7255753768720849630767399215373753335,
%U A270371 44436679763085787755205863082559307822924182270889047678247210478391618529
%N A270371 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/4,1/9,1/16,...).
%C A270371 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 A270371 See A269993 for a guide to related sequences.
%H A270371 Clark Kimberling, <a href="/A270371/b270371.txt">Table of n, a(n) for n = 1..13</a>
%H A270371 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270371 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270371 sqrt(1/2) = 1/2 + 1/(4*2) + 1/(9*2) + 1/(16*3) + 1/(25*7) + ...
%t A270371 r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
%t A270371 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270371 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270371 x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]
%o A270371 (PARI) r(k) = 1/k^2;
%o A270371 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270371 a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 21 2016
%Y A270371 Cf. A269993.
%K A270371 nonn,frac,easy
%O A270371 1,1
%A A270371 _Clark Kimberling_, Mar 20 2016