OFFSET
1,3
COMMENTS
Max Alekseyev (see link below) proved the following: suppose that N = c(1)F(1) - c(2)F(2) + c(3)F(3) - ..., where F(i) are Fibonacci numbers and each coefficient c(i) is either 0 or 1 with no adjacent unit coefficients. Then these coefficients are exactly those produced by the greedy algorithm: N = c(0)/x + c(1)/x^2 + c(2)/x^3 + ... . It follows that there are only finitely many nonzero terms and that the representation is unique for the stated properties.
c(0)=Floor(N*x) (as in A000201, the lower Wythoff sequence). Thus as N*x-c(0) is the fractional part {N*x} of N*x, we have {N*x} represented as a sum of finitely many fractions 1/x^k.
LINKS
Max Alekseyev, Re: Representations found by the greedy algorithm, SeqFan Mailing List, Dec 19 2006
EXAMPLE
First five rows:
1 1
3 0 0 1
4 1 0 1
6 0 1 0 0 1
8 0 0 0 0 1
Row 4 matches 6 = 6/x + 0/x^2 + 1/x^3 + 0/x^4 + 0/x^5 + 1/x^6.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Dec 23 2006
STATUS
approved