A079954 Partial sums of A030301.

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

Author

N. J. A. Sloane, Feb 22 2003

Keywords

• nonn

Links

• Kevin Ryde, Table of n, a(n) for n = 1..8192
• 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.

Crossrefs

Cf. A000523, A000975, A030301, A079947.

Programs

Magma

[&+[Floor(Log(k)/Log(2)) mod 2:k in [1..n]]:n in [1..75]]; // Marius A. Burtea, Oct 25 2019

Mathematica

Accumulate@ Flatten@ Table[1 - Mod[n, 2], {n, 7}, {2^(n - 1)}] (* Michael De Vlieger, Oct 29 2022 *)

PARI

a(n) = my(k=logint(n,2), p=(2<Kevin Ryde, Jul 23 2019 */

Python

def A079954(n): return ((1<Chai Wah Wu, Jan 30 2023

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