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a(0) = 0; a(2n) = 4*a(n), a(2n+1) = a(n) + 1.
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%I #11 Nov 28 2017 08:44:53

%S 0,1,4,2,16,5,8,3,64,17,20,6,32,9,12,4,256,65,68,18,80,21,24,7,128,33,

%T 36,10,48,13,16,5,1024,257,260,66,272,69,72,19,320,81,84,22,96,25,28,

%U 8,512,129,132,34,144,37,40,11,192,49,52,14,64,17,20,6,4096,1025,1028,258,1040

%N a(0) = 0; a(2n) = 4*a(n), a(2n+1) = a(n) + 1.

%H N. J. A. Sloane and J. A. Sellers, <a href="https://arxiv.org/abs/math/0312418">On non-squashing partitions</a>, arXiv:math/0312418 [math.CO], 2003.

%H N. J. A. Sloane and J. A. Sellers, <a href="https://doi.org/10.1016/j.disc.2004.11.014">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.

%p S := 4; f := proc(n) global S; option remember; if n=0 then RETURN(0); fi; if n mod 2 = 0 then RETURN(S*f(n/2)); else f((n-1)/2)+1; fi; end;

%t a[n_] := a[n] = Which[n == 0, 0, EvenQ[n], 4 a[n/2], True, a[(n-1)/2] + 1];

%t Table[a[n], {n, 0, 68}] (* _Jean-François Alcover_, Nov 28 2017 *)

%Y Cf. A087808, A090639.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Dec 14 2003