

A118017


Largest denominators in even Egyptian fraction representation of 1.


0




OFFSET

1,1


COMMENTS

An Egyptian Fraction representation of a rational number a/b is a list of distinct unit fractions with sum a/b. We will call it an even Egyptian Fraction representation if only even integers are used as denominators. The nth term of this sequence gives the largest denominators that appear in an Egyptian Fraction Representation of one with length n+3. For instance, 1/2 + 1/4 + 1/6 + 1/12 gives a 4term even representation of one, which is the shortest possible Egyptian even fraction representation of one.
This sequence can be derived from the Sylvester sequence (A000058). If s(n) represents the Sylvester sequence, s(n)1 is the largest denominator appearing in an n term Egyptian fraction representation of 1. There is a onetoone correspondence between kterm representations and (k+1)term even representation for k<12. An even representation has to have at least 4 terms, thus a(1) is related to s(3). a(1) = 2*(s(3)  1), etc.
Excluding the first two terms of the Pythagorean spiral sequence (A053631) yields this sequence.


REFERENCES

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. 340342


LINKS

Table of n, a(n) for n=1..5.


FORMULA

a(n+1) = 1/2*a(n)^2 + a(n).


EXAMPLE

a(2) = 1/2*12^2 + 12 so a(2) = 84.


CROSSREFS

Cf. A000058, A007018, A053631.
A053631 and A127690 are very similar to this sequence.
Sequence in context: A003209 A155645 A095267 * A225785 A098206 A104911
Adjacent sequences: A118014 A118015 A118016 * A118018 A118019 A118020


KEYWORD

nonn


AUTHOR

Teena Carroll, Jul 06 2011


STATUS

approved



