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A270373 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1, 1/4, 1/9, 1/16, ...). 1

%I #15 Feb 24 2018 10:10:56

%S 3,4,7,25,5546,36482088,14423934280776257,

%T 1969937215073991451613042447271867,

%U 3160555685801520768089757205744771458914199650397475324265981061618

%N Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, 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="/A270373/b270373.txt">Table of n, a(n) for n = 1..12</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(2) - 1 = 1/3 + 1/(4*4) + 1/(9*7) + 1/(16*25) + ...

%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[2] - 1; 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(2)-1) = 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

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Last modified May 25 16:58 EDT 2024. Contains 372801 sequences. (Running on oeis4.)