A179180 Partial sums of A007895.

0, 1, 2, 3, 5, 6, 8, 10, 11, 13, 15, 17, 20, 21, 23, 25, 27, 30, 32, 35, 38, 39, 41, 43, 45, 48, 50, 53, 56, 58, 61, 64, 67, 71, 72, 74, 76, 78, 81, 83, 86, 89, 91, 94, 97, 100, 104, 106, 109, 112, 115, 119, 122, 126, 130, 131, 133, 135, 137, 140, 142, 145

Offset: 0

Author

Walt Rorie-Baety, Jun 30 2010

Keywords

• nonn

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

Examples

			For n = 6, a(n) = 1+1+1+2+1+2 = 8.
		

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

Crossrefs

Cf. A001622, A007895, A214979.

Programs

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 *)

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

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