A123753 Partial sums of A070941.

1, 3, 6, 9, 13, 17, 21, 25, 30, 35, 40, 45, 50, 55, 60, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343

Offset

0,2

Links

Peter Luschny, Table of n, a(n) for n = 0..10000

Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.

Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47.

Formula

a(n) = A003314(n+1)+1. - Reinhard Zumkeller, Oct 12 2006

Let bil(n) = floor(log_2(n)) + 1 for n>0, bil(0) = 0 and b(n) = n + n*bil(n) - 2^bil(n) + 1 then a(n) = b(n+1). (This suggests that '0' be prepended to this sequence.) - Peter Luschny, Dec 02 2017

Maple

A123753 := proc(n) local i, J, z; i := n+1: J := i; i := i-1; z := 1;
while 0 <= i do J := J+i; i := i-z; z := z+z od; J end:
seq(A123753(n), n=0..57); # Peter Luschny, Nov 30 2017
# Alternatively:
a := n -> (n+1)*(1 + ilog2(2*n+3)) - 2^ilog2(2*n+3) + 1:
seq(a(n), n=0..57); # Peter Luschny, Dec 02 2017

Mathematica

a[n_] := (n + 1)(1 + IntegerLength[n + 1, 2]) - 2^IntegerLength[n + 1, 2] + 1;
Table[a[n], {n, 0, 57}] (* Peter Luschny, Dec 02 2017 *)

Prog

(Python)
def A123753(n):
    s, i, z = n+1, n, 1
    while 0 <= i: s += i; i -= z; z += z
    return s
print([A123753(n) for n in range(0, 58)]) # Peter Luschny, Nov 30 2017
(Python)
def A123753(n): return (n+1)*(1+(m:=n.bit_length()))-(1<<m)+1 # Chai Wah Wu, Mar 29 2023

Crossrefs

Cf. A001855, A003314, A033156, A054248, A061168, A083652, A097383, A295508.

Sequence in context: A278449 A006590 A061781 * A124288 A256966 A280944

Adjacent sequences: A123750 A123751 A123752 * A123754 A123755 A123756

Keyword

nonn

Author

Reinhard Zumkeller, Oct 12 2006

Status

approved