

A270519


Denominators of rEgyptian fraction expansion for sqrt(2)  1, where r(k) = 1/k!.


0




OFFSET

1,1


COMMENTS

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(k1)), and f(k) = f(k1)  r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the rEgyptian fraction for x.
See A269993 for a guide to related sequences.


LINKS



EXAMPLE

sqrt(2)  1 = 1/(1*3) + 1/(2*7) + 1/(6*18) + 1/(24*217) + ...


MATHEMATICA

r[k_] := 1/k!; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k  1]]
f[x_, k_] := f[x, k] = f[x, k  1]  r[k]/n[x, k]
x = Sqrt[2]  1; Table[n[x, k], {k, 1, z}]


PROG

(PARI) r(k) = 1/k!;
f(k, x) = if (k==0, x, f(k1, x)  r(k)/a(k, x); );
a(k, x=sqrt(2)1) = ceil(r(k)/f(k1, x)); \\ Michel Marcus, Mar 31 2016


CROSSREFS



KEYWORD

nonn,frac,easy


AUTHOR



STATUS

approved



