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A270350 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1, 1/2, 1/4, 1/8, ...) 1
2, 3, 4, 44, 1446, 3423518, 263631451737996, 70985515555913904515293113895, 8645798497265822420998718966216306501746531100894289290802, 78713180847550502513757221862401308079612732875925186430170968601702893264445327722349352410275677392885249561650440 (list; graph; refs; listen; history; text; internal format)
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(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.

See A269993 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..12

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

EXAMPLE

sqrt(3) - 1 = 1/2 + 1/(2*3) + 1/(4*4) + ...

MATHEMATICA

r[k_] := 2/2^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[3] - 1; Table[n[x, k], {k, 1, z}]

PROG

(PARI) r(k) = 2/2^k;

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

a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 18 2016

CROSSREFS

Cf. A269993.

Sequence in context: A024634 A134305 A000033 * A060411 A037322 A037429

Adjacent sequences:  A270347 A270348 A270349 * A270351 A270352 A270353

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Mar 17 2016

STATUS

approved

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Last modified October 20 07:17 EDT 2018. Contains 316378 sequences. (Running on oeis4.)