OFFSET
0,4
COMMENTS
Every nonnegative integer has a unique Zeckendorf representation (A014417) as a sum of nonconsecutive Fibonacci numbers F_{k}, with k>1. Likewise, every integer has a unique NegaFibonacci representation as a sum of nonconsecutive F_{-k}, with k>0 (A215022 for the positive integers, A215023 for the negative). So the F_{k} summing to n are transformed to F_{-k+1} and summed. Since the representations are unique and mapped one-to-one, every integer appears exactly once in the sequence.
a(n) changes sign at each Fibonacci number, since NegaFibonacci representations with an odd number of fibits are positive and those with an even number are negative.
REFERENCES
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 169.
E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.
LINKS
Garth A. T. Rose, Table of n, a(n) for n = 0..1000
Sean A. Irvine, Java program (github)
Wikipedia, Zeckendorf's theorem
EXAMPLE
10 is 8 + 2, or F_6 + F_3. a(10) is then F_{-5} + F_{-2} = 5 + (-1) = 4.
PROG
(Sage)
def a309076(n):
result = 0
lnphi = ln((1+sqrt(5))/2)
while n > 0:
k = floor(ln(n*sqrt(5)+1/2)/lnphi)
n = n - fibonacci(k)
result = result + fibonacci(1 - k)
return result
CROSSREFS
KEYWORD
base,sign
AUTHOR
Garth A. T. Rose, Jul 10 2019
STATUS
approved