A309076 The Zeckendorf representation of n read as a NegaFibonacci representation.

0, 1, -1, 2, 3, -3, -2, -4, 5, 6, 4, 7, 8, -8, -7, -9, -6, -5, -11, -10, -12, 13, 14, 12, 15, 16, 10, 11, 9, 18, 19, 17, 20, 21, -21, -20, -22, -19, -18, -24, -23, -25, -16, -15, -17, -14, -13, -29, -28, -30, -27, -26, -32, -31, -33

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

Cf. A000045, A014417, A215022, A215023.

Sequence in context: A265339 A182865 A131469 * A343326 A073078 A360296

Adjacent sequences: A309073 A309074 A309075 * A309077 A309078 A309079

Keyword

base,sign

Author

Garth A. T. Rose, Jul 10 2019

Status

approved