login
A143157
Partial sums of A091512.
4
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
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
a(n) ~ n^2. - Amiram Eldar, Sep 10 2024
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
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jul 27 2008
EXTENSIONS
a(0) = 0 prepended and more terms computed by Antti Karttunen, Oct 25 2014
STATUS
approved