|
Search: id:A104324
|
|
|
| A104324 |
|
Number of runs (of equal bits) in the Zeckendorf (binary) representation of n. |
|
+0
4
|
|
| 1, 2, 2, 3, 2, 3, 4, 2, 3, 4, 4, 5, 2, 3, 4, 4, 5, 4, 5, 6, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 6, 7, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 6, 7, 4, 5, 6, 6, 7, 6, 7, 8, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 6, 7, 4, 5, 6, 6, 7, 6, 7, 8, 4, 5, 6, 6, 7, 6, 7, 8, 6, 7, 8, 8, 9, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 6, 7, 4, 5, 6, 6
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
Series has some interesting fractal properties (plot it!)
First occurrence of k is: 1,2,4,7,12,20,33,54, ..., A000071(k+1): Fibonacci numbers - 1. - Robert G. Wilson v, Apr 25 2006.
|
|
REFERENCES
|
E. Zeckendorf, Representation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liege 41, 179-182, 1972.
|
|
LINKS
|
R. Knott Using Fibonacci Numbers to Represent Whole Numbers
|
|
EXAMPLE
|
14 = 13+1 as a sum of Fibonacci numbers = 100001(in Fibonacci base) using the least number of 1's (Zeckendorf Rep): it consists of 3 runs: one 1, four 0's, one 1 so a(14)=3
|
|
MAPLE
|
with(combinat, fibonacci):fib:=fibonacci: zeckrep:=proc(N)local i, z, j, n; i:=2; z:=NULL; n:=N; while fib(i)<=n do i:=i+1 od; print(i=fib(i)); for j from i-1 by -1 to 2 do if n>=fib(j) then z:=z, 1; n:=n-fib(j) else z:=z, 0 fi od; [z] end proc: countruns:=proc(s)local i, c, elt; elt:=s[1]; c:=1; for i from 2 to nops(s) do if s[i]<>s[i-1] then c:=c+1 fi od; c end proc: seq(countruns(zeckrep(n)), n=1..100);
|
|
MATHEMATICA
|
f[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; While[ fr[[1]] == 0, fr = Rest@fr]; Length@ Split@ fr]; Array[f, 105] (from Robert G. Wilson v (rgwv(at)rgwv.com), Apr 25 2006)
|
|
CROSSREFS
|
Cf. A014417, A104325.
Sequence in context: A046773 A101037 A002199 this_sequence A131818 A070081 A034883
Adjacent sequences: A104321 A104322 A104323 this_sequence A104325 A104326 A104327
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Ron Knott (enquiry(AT)ronknott.com), Mar 01 2005
|
|
|
Search completed in 0.002 seconds
|
