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
Examples
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.
Links
- Antti Karttunen, Table of n, a(n) for n = 0..8191
Programs
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Mathematica
{0}~Join~Accumulate@ Array[IntegerExponent[(2 #)^#, 2] &, 56] (* Michael De Vlieger, Sep 29 2019 *) -
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
Formula
Row sums of triangle A143156.
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
Extensions
a(0) = 0 prepended and more terms computed by Antti Karttunen, Oct 25 2014