A133775 Number of 0's in the minimal "phinary" (A130600) representation of n.
0, 2, 3, 2, 5, 5, 7, 6, 5, 5, 4, 8, 8, 8, 7, 8, 8, 11, 10, 9, 9, 8, 8, 8, 9, 8, 7, 7, 6, 11, 11, 11, 10, 11, 11, 12, 11, 10, 10, 9, 11, 11, 11, 10, 11, 11, 15, 14, 13, 13, 12, 12, 12, 13, 12, 11, 11, 10, 11, 11, 11, 10, 11, 11, 13, 12, 11, 11, 10, 10, 10, 11, 10, 9, 9, 8, 14, 14, 14, 13, 14, 14
Offset: 1
Keywords
Examples
A130600(5)=10001001, which has five 0's. So a(5)=5.
References
- Zeckendorf, E., 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
- Casey Mongoven and T. D. Noe, Table of n, a(n) for n = 1..1000
- Ron Knott, Using Powers of Phi to represent Integers.
Programs
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Mathematica
nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; last1 = Position[d[[1]], 1][[-1, 1]]; Count[Take[d[[1]], last1], 0], {n, 1, nn}] (* T. D. Noe, May 20 2011 *)
Formula
For n > 1, a(n) <= A190796(n) - 2. - Charles R Greathouse IV, Apr 21 2023