%I M2448 #28 Feb 28 2019 11:47:15

%S 0,1,3,5,9,12,18,21,29,34,44,48,60,67,81,85,101,110,128,134,154,165,

%T 187,192,216,229,255,263,291,306,336,341,373,390,424,434,470,489,527,

%U 534,574,595,637,649,693,716,762,768,816,841,891,905,957,984,1038

%N a(n) = cost of minimal multiplication-cost addition chain for n.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005766/b005766.txt">Table of n, a(n) for n = 1..1000</a>

%H J.-P. Allouche and J. Shallit, <a href="https://doi.org/10.1016/0304-3975(92)90001-V">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197, ex. 21.

%H R. L. Graham, A. C.-C. Yao, F. F. Yao, <a href="http://dx.doi.org/10.1016/0012-365X(78)90111-5">Addition chains with multiplicative cost</a> Discrete Math. 23 (1978), 115-119.

%H R. L. Graham et al., <a href="/A005766/a005766.pdf">Addition chains with multiplicative cost</a> [Cached copy]

%H R. Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>

%H R. Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>

%H <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>

%F a(2n)=a(n)+n^2, a(2n+1)=a(n)+n(n+2). - _Ralf Stephan_, May 04 2003

%F G.f.: 1/(1-x) * sum(k>=0, x^2^(k+1)(1+2x^2^k-x^2^(k+1))/(1-x^2^(k+1))^2). - _Ralf Stephan_, Jul 27 2003

%F a(n) = sum(k=1, n, A007814(n) + 2*A025480(n-1)). - _Ralf Stephan_, Oct 30 2003

%t a[n_] := Sum[v = IntegerExponent[k, 2]; v + k/2^v - 1, {k, 1, n}];

%t Array[a, 55] (* _Jean-François Alcover_, Feb 28 2019 *)

%o (PARI) a(n)=if(n<1,0,if(n%2==0,a(n/2)+n^2/4,a((n-1)/2)+(n-1)*(n+3)/4))

%o (PARI) a(n)=sum(k=1,n,valuation(k,2)+k/2^valuation(k,2)-1)

%Y Partial sums of A089265.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, _Jeffrey Shallit_, _Robert G. Wilson v_

%E More terms from _Ralf Stephan_, May 04 2003