A295508 -id:A295508 - cp's OEIS frontend

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

Reinhard Zumkeller, Oct 12 2006

nonn

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.

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

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

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 *)

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

def A123753(n): return (n+1)*(1+(m:=n.bit_length()))-(1<Chai Wah Wu, Mar 29 2023

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

A295513 a(n) = n*bil(n) - 2^bil(n) where bil(0) = 0 and bil(n) = floor(log_2(n)) + 1 for n>0.

Original entry on oeis.org

-1, -1, 0, 2, 4, 7, 10, 13, 16, 20, 24, 28, 32, 36, 40, 44, 48, 53, 58, 63, 68, 73, 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278

Offset: 0

Peter Luschny, Dec 02 2017

sign

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.

Cf. A001855, A003314, A033156, A054248, A061168, A083652, A097383, A123753, A295508.

A295513 := proc(n) local i, s, z; s := -1; i := n-1; z := 1;
while 0 <= i do s := s+i; i := i-z; z := z+z od; s end:
seq(A295513(n), n=0..57);

a[n_] := n IntegerLength[n, 2] - 2^IntegerLength[n, 2];
Table[a[n], {n, 0, 58}]

def A295513(n): return n*(m:=(n-1).bit_length())-(1<Chai Wah Wu, Mar 29 2023

A001855(n) = a(n) + 1.
A033156(n) = a(n) + 2n.
A003314(n) = a(n) + n.
A083652(n) = a(n+1) + 2.
A061168(n) = a(n+1) - n + 1.
A123753(n) = a(n+1) + n + 2.
A097383(n) = a(n+1) - div(n-1, 2).
A054248(n) = a(n) + n + rem(n, 2).

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A123753 Partial sums of A070941.

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A295513 a(n) = n*bil(n) - 2^bil(n) where bil(0) = 0 and bil(n) = floor(log_2(n)) + 1 for n>0.

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