The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A126723 Successive rows of coefficients c(0), c(1), c(2),... for the greedy-algorithm representation of a positive integer n: n = c(0)/x + c(1)/x^2 + c(2)/x^3 + ..., where x = (1+sqrt(5))/2. 1
 1, 1, 3, 0, 0, 1, 4, 1, 0, 1, 6, 0, 1, 0, 0, 1, 8, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 1, 11, 0, 0, 1, 0, 1, 12, 1, 0, 1, 0, 1, 14, 0, 1, 0, 1, 0, 0, 1, 16, 0, 0, 0, 1, 0, 0, 1, 17, 1, 0, 0, 1, 0, 0, 1, 19, 0, 1, 0, 0, 0, 0, 1, 21, 0, 0, 0, 0, 0, 0, 1, 22, 1, 0, 0, 0, 0, 0, 1, 24, 0, 0, 1, 0, 0, 0, 1, 25, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000045, A000201. Sequence in context: A170840 A300725 A035630 * A325846 A325735 A235794 Adjacent sequences: A126720 A126721 A126722 * A126724 A126725 A126726 KEYWORD nonn,tabf AUTHOR Clark Kimberling, Dec 23 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 29 12:22 EDT 2023. Contains 361599 sequences. (Running on oeis4.)