A020899 Numbers k with an odd number of terms in their Zeckendorf representation (write k as a sum of non-consecutive distinct Fibonacci numbers).

1, 2, 3, 5, 8, 12, 13, 17, 19, 20, 21, 25, 27, 28, 30, 31, 32, 34, 38, 40, 41, 43, 44, 45, 48, 49, 50, 52, 55, 59, 61, 62, 64, 65, 66, 69, 70, 71, 73, 77, 78, 79, 81, 84, 88, 89, 93, 95, 96, 98, 99, 100, 103, 104, 105, 107, 111, 112, 113, 115, 118, 122, 124, 125

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Numbers k such that A095076(k) = 1. - Amiram Eldar, Feb 05 2023

References

• C. G. Lekkerkerker, Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci, Simon Stevin 29 (1952), 190-195.
• Edouard Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41 (1972), 179-182.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• D. E. Daykin, Representation of natural numbers as sums of generalized Fibonacci numbers, J. London Math. Soc. 35 (1960), 143-160.

Crossrefs

Cf. A007895, A014417, A095076.

Programs

Haskell

a020899 n = a020899_list !! (n-1)
a020899_list = filter (odd . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

Mathematica

Flatten @ Position[Mod[DigitCount[Select[Range[0, 1000], BitAnd[#, 2 #] == 0 &], 2, 1], 2], 1] - 1 (* Amiram Eldar, Feb 05 2023 *)

Formula

A007895(a(n)) mod 2 = 1. - Reinhard Zumkeller, Mar 10 2013

Extensions

Offset corrected by Reinhard Zumkeller, Mar 10 2013