A143157 Partial sums of A091512.

0, 1, 5, 8, 20, 25, 37, 44, 76, 85, 105, 116, 152, 165, 193, 208, 288, 305, 341, 360, 420, 441, 485, 508, 604, 629, 681, 708, 792, 821, 881, 912, 1104, 1137, 1205, 1240, 1348, 1385, 1461, 1500, 1660, 1701, 1785, 1828, 1960, 2005, 2097, 2144, 2384, 2433, 2533, 2584, 2740, 2793, 2901, 2956, 3180

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..8191

Formula

Partial sums of A091512 = Sum_{j>=1} j*A001511(j), where A001511 is the ruler sequence.

Row sums of triangle A143156.

a(n) = A249152(2*n)/2 = A249153(n) / 2. - Antti Karttunen, Oct 25 2014

a(n) = (1/2)*n*(n + 1) + Sum_{i=1..n} i*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Sep 03 2019; Jan 22 2021

G.f. A(x) satisfies: A(x) = 2*A(x^2)*(1 + x) + x/(1 - x)^3. - Ilya Gutkovskiy, Oct 30 2019

Example

a(4) = 20 = sum of row 4 terms of triangle A143156, (7 + 6 + 4 + 3).
a(4) = 20 = partial sums of first 4 terms of A091512: (1 + 4 + 3 + 12).
a(4) = 20 = Sum_{j=1..4} j*A001511(j) = 1*1 + 2*2 + 3*1 + 4*3.

Mathematica

{0}~Join~Accumulate@ Array[IntegerExponent[(2 #)^#, 2] &, 56] (* Michael De Vlieger, Sep 29 2019 *)

Prog

(Python)
def A143157(n): return sum(i*(~i&i-1).bit_length() for i in range(2, 2*n+1, 2))>>1 # Chai Wah Wu, Jul 11 2022

Crossrefs

Cf. A001511, A007814, A091512, A143156, A249152, A249153.

Sequence in context: A270321 A271885 A295901 * A275236 A270022 A271195

Adjacent sequences: A143154 A143155 A143156 * A143158 A143159 A143160

Keyword

nonn

Author

Gary W. Adamson, Jul 27 2008

Extensions

a(0) = 0 prepended and more terms computed by Antti Karttunen, Oct 25 2014

Status

approved