A049909 a(n) = a(1) + a(2) + ... + a(n-1) - a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the smallest integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 3.
1, 2, 3, 4, 9, 15, 31, 63, 127, 192, 416, 848, 1702, 3409, 6819, 13639, 27279, 40920, 88660, 180730, 363167, 727188, 1454808, 2909840, 5819745, 11639554, 23279140, 46558296, 93116598, 186233201, 372466403, 744932807, 1489865615
Offset: 1
Keywords
Examples
From _Petros Hadjicostas_, Nov 07 2019: (Start)
a(4) = -a(2^ceiling(log_2(4-1)) + 2 - 4) + a(1) + a(2) + a(3) = -a(2) + a(1) + a(2) + a(3) = 4.
a(5) = -a(2^ceiling(log_2(5-1)) + 2 - 5) + a(1) + a(2) + a(3) + a(4) = -a(1) + a(1) + a(2) + a(3) + a(4) = 9.
a(6) = -a(2^ceiling(log_2(6-1)) + 2 - 6) + a(1) + a(2) + a(3) + a(4) + a(5) = -a(4) + a(1) + a(2) + a(3) + a(4) + a(5) = 15.
a(7) = -a(7 - 1 - A006257(7-2)) + Sum_{i = 1..6} a(i) = -a(3) + Sum_{i = 1..6} a(i) = 31.
a(8) = -a(8 - 1 - A006257(8-2)) + Sum_{i = 1..7} a(i) = -a(2) + Sum_{i = 1..7} a(i) = 63. (End)
Programs
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Maple
s:= proc(n) option remember; `if`(n<1, 0, a(n)+s(n-1)) end: a:= proc(n) option remember; `if`(n<4, [1, 2, 3][n], s(n-1) - a(Bits:-Iff(n-2$2) + 3 - n)) end: seq(a(n), n=1..40); # Petros Hadjicostas, Nov 07 2019
Formula
From Petros Hadjicostas, Nov 07 2019: (Start)
a(n) = -a(2^ceiling(log_2(n-1)) + 2 - n) + Sum_{i = 1..n-1} a(i) for n >= 4.
a(n) = -a(n - 1 - A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4. (End)
Extensions
Name edited by Petros Hadjicostas, Nov 07 2019