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A079954
Partial sums of A030301.
3
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, 10, 10, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 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
OFFSET
1,3
LINKS
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 49-50.
FORMULA
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
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
MATHEMATICA
Accumulate@ Flatten@ Table[1 - Mod[n, 2], {n, 7}, {2^(n - 1)}] (* Michael De Vlieger, Oct 29 2022 *)
PROG
(PARI) a(n) = my(k=logint(n, 2), p=(2<<k)\3); if(bittest(k, 0), n-p, p); /* Kevin Ryde, Jul 23 2019 */
(Magma) [&+[Floor(Log(k)/Log(2)) mod 2:k in [1..n]]:n in [1..75]]; // Marius A. Burtea, Oct 25 2019
(Python)
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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 22 2003
STATUS
approved