A079954 - OEIS

%I #30 Jan 30 2023 19:34:26

%S 0,1,2,2,2,2,2,3,4,5,6,7,8,9,10,10,10,10,10,10,10,10,10,10,10,10,10,

%T 10,10,10,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,

%U 30,31,32,33,34,35,36,37,38,39,40,41,42,42,42,42,42,42,42,42,42,42,42,42,42,42

%N Partial sums of A030301.

%H Kevin Ryde, <a href="/A079954/b079954.txt">Table of n, a(n) for n = 1..8192</a>

%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, pp. 49-50.

%F a(n) = (n - 1 - (2/3)*(4^e_4-1) - (-1)^e_2*(n - 1 - 2*(4^e_4-1)))/2 where e_4 = floor(log_4(n)) and e_2 = floor(log_2(n)) = floor(log_4(n^2)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003

%F a(n) = n - A079947(n). Let k=A000523(n), then a(n) = A000975(k) if k even, or a(n) = n - A000975(k) if k odd. - _Kevin Ryde_, Jul 23 2019

%t Accumulate@ Flatten@ Table[1 - Mod[n, 2], {n, 7}, {2^(n - 1)}] (* _Michael De Vlieger_, Oct 29 2022 *)

%o (PARI) a(n) = my(k=logint(n,2), p=(2<<k)\3); if(bittest(k,0), n-p, p); /* _Kevin Ryde_, Jul 23 2019 */

%o (Magma) [&+[Floor(Log(k)/Log(2)) mod 2:k in [1..n]]:n in [1..75]]; // _Marius A. Burtea_, Oct 25 2019

%o (Python)

%o def A079954(n): return ((1<<k)-2)//3 if (k:=n.bit_length())&1 else n-((1<<k)-1)//3 # _Chai Wah Wu_, Jan 30 2023

%Y Cf. A000523, A000975, A030301, A079947.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Feb 22 2003