A179180 - OEIS

%I #15 Dec 09 2021 08:37:17

%S 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,

%T 48,50,53,56,58,61,64,67,71,72,74,76,78,81,83,86,89,91,94,97,100,104,

%U 106,109,112,115,119,122,126,130,131,133,135,137,140,142,145

%N Partial sums of A007895.

%C 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

%H Clark Kimberling, <a href="/A179180/b179180.txt">Table of n, a(n) for n = 0..10000</a>

%H Christian Ballot, <a href="https://www.fq.math.ca/Papers1/51-4/BallotZeckendorfBase.pdf">On Zeckendorf and Base b Digit Sums</a>, Fibonacci Quarterly, Vol. 51, No. 4 (2013), pp. 319-325.

%H Vaclav Kotesovec, <a href="/A179180/a179180.jpg">Graph - the asymptotic ratio</a>

%F 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

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

%t s = Reverse[Table[Fibonacci[n + 1], {n, 1, 70}]];

%t 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}];

%t v1 = Table[v[n], {n, 1, z}]

%t (* _Peter J. C. Moses_, Oct 18 2012 *)

%t DigitCount[Select[Range[0, 500], BitAnd[#, 2*#] == 0&], 2, 1] // Accumulate (* _Jean-François Alcover_, Jan 25 2018 *)

%Y Cf. A001622, A007895, A214979.

%K nonn

%O 0,3

%A _Walt Rorie-Baety_, Jun 30 2010

%E Corrected term a(17); the working list of the terms were not in order. _Walt Rorie-Baety_, Jun 30 2010

%E Extended by _Clark Kimberling_, Oct 23 2012