A117116 Denominators of an Egyptian Fraction for phi = (1+sqrt(5))/2.

1, 2, 9, 145, 37986, 2345721887, 26943815937041299094, 811625643619814151937413504618770581764, 697120590223140234675813998970770820981012350673738243594006422610850113672220

Offset

0,2

Comments

For each term, the largest possible unit fraction is used.

Links

Amiram Eldar, Table of n, a(n) for n = 0..11

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342.

David Eppstein, Algorithms for Egyptian Fractions.

Eric Weisstein's World of Mathematics, Egyptian Fraction.

Index entries for sequences related to Egyptian fractions.

Example

a(4) = 145 because 1/145 is the largest unit fraction less than phi - 1/1 - 1/2 - 1/9.

Maple

v[0]:=1: for n from 1 to 10 do x:=ceil(1/((1+sqrt(5))/2-add(1/v[i], i=0..n-1))); while not x::integer do Digits:=2*Digits; x:=ceil(1/((1+sqrt(5))/2-add(1/v[i], i=0..n-1))) od; v[n]:=x; od: seq(v[i], i=0..8);  # Paolo P. Lava, May 03 2018

Mathematica

a = {1}; k = N[(Sqrt[5] - 1)/2, 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a (* Artur Jasinski, Sep 22 2008 *)

Prog

(PARI) x = (1 + sqrt(5))/2 - 1;
f(x, k) = if(k<1, x, f(x, k - 1) - 1/n(x, k));
n(x, k) = ceil(1/f(x, k - 1));
for(k = 0, 9, print1(if(k==0, 1, n(x, k)), ", ")) \\ Indranil Ghosh, Mar 27 2017

Crossrefs

Cf. A001622.

Sequence in context: A376321 A174954 A193440 * A211935 A133468 A182948

Adjacent sequences: A117113 A117114 A117115 * A117117 A117118 A117119

Keyword

nonn,frac

Author

Paolo P. Lava and Giorgio Balzarotti, Apr 19 2006

Extensions

Edited by Don Reble, Apr 21 2006

Status

approved