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The sequence is defined as $a(1)=1,a(n)=a(n-1)+a\left(\left\lfloor \log _2(n)\right\rfloor \right)$

A natural way do this is

ClearAll[a];
a[1] = 1;
a[n_] := a[n] = a[n - 1] + a[Floor@Log2@N@n];
Table[a[i], {i, 1, 2^20}]; // AbsoluteTiming

It's not very quickly enough. I think there is a iteration solution using Nest or Fold,but I can't get it.

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  • $\begingroup$ You can get a slight improvement by using BitLength@n-1 instead of Floor@Log2@N@n $\endgroup$ – Lukas Lang Aug 25 '17 at 7:29
  • $\begingroup$ Do you need to generate the whole sequence from 1 to some N, or do you want a faster way to generate results for some large distinct arguments? $\endgroup$ – ciao Aug 25 '17 at 7:31
  • $\begingroup$ @ciao I need to generate the whole sequence from 1 to N. $\endgroup$ – mathe Aug 25 '17 at 8:12
  • $\begingroup$ @mathe - then see my answer. $\endgroup$ – ciao Aug 25 '17 at 8:13
  • $\begingroup$ @mathe - did answer address issue ? $\endgroup$ – ciao Aug 26 '17 at 2:17
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This will be much much faster:

a3[1]={1};
a3[m_] := Module[{t = 2^Range[Floor[Log2[m]]], a},

   a[1] = 1;
   a[n_] := a[n] = a[n - 1] + a[Floor[Log2[n]]];

   t[[-1]] = m - Tr[Most@t] - 1;
   Accumulate@Prepend[Join @@ MapThread[ConstantArray, {a /@ Range[Length@t], t}], 1]];

Using:

a3[X]

will produce the same output as 

Table[a[i], {i, 1, X}]
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