A270371 - OEIS

%I #14 Nov 16 2024 23:20:58

%S 2,2,2,3,7,7702,1234163819,1590823281229385753,

%T 7255753768720849630767399215373753335,

%U 44436679763085787755205863082559307822924182270889047678247210478391618529

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

%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="/A270371/b270371.txt">Table of n, a(n) for n = 1..13</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 sqrt(1/2) = 1/2 + 1/(4*2) + 1/(9*2) + 1/(16*3) + 1/(25*7) + ...

%t r[k_] := 1/k^2; 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 = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]

%o (PARI) r(k) = 1/k^2;

%o f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););

%o a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 21 2016

%Y Cf. A269993.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Mar 20 2016