A122247 Partial sums of A005187.

0, 1, 4, 8, 15, 23, 33, 44, 59, 75, 93, 112, 134, 157, 182, 208, 239, 271, 305, 340, 378, 417, 458, 500, 546, 593, 642, 692, 745, 799, 855, 912, 975, 1039, 1105, 1172, 1242, 1313, 1386, 1460, 1538, 1617, 1698, 1780, 1865, 1951, 2039, 2128, 2222, 2317, 2414

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi 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. Also first authors' copy, 2016. See example 6.2.

Formula

a(n) = Sum_{k=1..n} Sum_{j=0..n} floor(k/2^j).

G.f.: (1/(1-x))*Sum_{k>=0} x^(2^k)/((1-x)*(1-x^(2^k))).

Maple

b:= proc(n) option remember; `if`(n=0, 0, b(iquo(n, 2))+n) end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)+a(n-1)) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 25 2022

Mathematica

Accumulate[Table[2n-Count[IntegerDigits[2 n, 2], 1], {n, 0, 70}]] (* Harvey P. Dale, Oct 22 2011 *)

Crossrefs

Cf. A005187 (first differences).

Sequence in context: A102216 A001182 A264599 * A126255 A267682 A194804

Adjacent sequences: A122244 A122245 A122246 * A122248 A122249 A122250

Keyword

easy,nonn

Author

Paul Barry, Aug 27 2006

Status

approved