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A179180 Partial sums of A007895.

%I A179180 #16 Jan 05 2025 19:51:39
%S A179180 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 A179180 48,50,53,56,58,61,64,67,71,72,74,76,78,81,83,86,89,91,94,97,100,104,
%U A179180 106,109,112,115,119,122,126,130,131,133,135,137,140,142,145
%N A179180 Partial sums of A007895.
%C A179180 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 A179180 Clark Kimberling, <a href="/A179180/b179180.txt">Table of n, a(n) for n = 0..10000</a>
%H A179180 Christian Ballot, <a href="https://web.archive.org/web/2024*/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 A179180 Vaclav Kotesovec, <a href="/A179180/a179180.jpg">Graph - the asymptotic ratio</a>
%F A179180 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 A179180 For n = 6, a(n) = 1+1+1+2+1+2 = 8.
%t A179180 s = Reverse[Table[Fibonacci[n + 1], {n, 1, 70}]];
%t A179180 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 A179180 v1 = Table[v[n], {n, 1, z}]
%t A179180 (* _Peter J. C. Moses_, Oct 18 2012 *)
%t A179180 DigitCount[Select[Range[0, 500], BitAnd[#, 2*#] == 0&], 2, 1] // Accumulate (* _Jean-François Alcover_, Jan 25 2018 *)
%Y A179180 Cf. A001622, A007895, A214979.
%K A179180 nonn
%O A179180 0,3
%A A179180 _Walt Rorie-Baety_, Jun 30 2010
%E A179180 Corrected term a(17); the working list of the terms were not in order. _Walt Rorie-Baety_, Jun 30 2010
%E A179180 Extended by _Clark Kimberling_, Oct 23 2012