OFFSET
0,3
COMMENTS
Total number of summands in Zeckendorf representations of all the numbers 1,2,...,n (for n>0); see the conjecture at A214979. - Clark Kimberling, Oct 23 2012
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..10000
Christian Ballot, On Zeckendorf and Base b Digit Sums, Fibonacci Quarterly, Vol. 51, No. 4 (2013), pp. 319-325.
Vaclav Kotesovec, Graph - the asymptotic ratio
FORMULA
a(n) ~ c * n * log(n), where c = (phi-1)/(sqrt(5)*log(phi)) = 0.574369... and phi is the golden ratio (A001622) (Ballot, 2013). - Amiram Eldar, Dec 09 2021
EXAMPLE
For n = 6, a(n) = 1+1+1+2+1+2 = 8.
MATHEMATICA
s = Reverse[Table[Fibonacci[n + 1], {n, 1, 70}]];
t2 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]][[2, 1]], # > 0 &]] &, Range[z]]; v[n_] := Sum[t2[[k]], {k, 1, n}];
v1 = Table[v[n], {n, 1, z}]
(* Peter J. C. Moses, Oct 18 2012 *)
DigitCount[Select[Range[0, 500], BitAnd[#, 2*#] == 0&], 2, 1] // Accumulate (* Jean-François Alcover, Jan 25 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Walt Rorie-Baety, Jun 30 2010
EXTENSIONS
Corrected term a(17); the working list of the terms were not in order. Walt Rorie-Baety, Jun 30 2010
Extended by Clark Kimberling, Oct 23 2012
STATUS
approved