This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143157 #49 Sep 10 2024 02:58:42
%S A143157 0,1,5,8,20,25,37,44,76,85,105,116,152,165,193,208,288,305,341,360,
%T A143157 420,441,485,508,604,629,681,708,792,821,881,912,1104,1137,1205,1240,
%U A143157 1348,1385,1461,1500,1660,1701,1785,1828,1960,2005,2097,2144,2384,2433,2533,2584,2740,2793,2901,2956,3180
%N A143157 Partial sums of A091512.
%H A143157 Antti Karttunen, <a href="/A143157/b143157.txt">Table of n, a(n) for n = 0..8191</a>
%F A143157 Partial sums of A091512 = Sum_{j>=1} j*A001511(j), where A001511 is the ruler sequence.
%F A143157 Row sums of triangle A143156.
%F A143157 a(n) = A249152(2*n)/2 = A249153(n) / 2. - _Antti Karttunen_, Oct 25 2014
%F A143157 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
%F A143157 G.f. A(x) satisfies: A(x) = 2*A(x^2)*(1 + x) + x/(1 - x)^3. - _Ilya Gutkovskiy_, Oct 30 2019
%F A143157 a(n) ~ n^2. - _Amiram Eldar_, Sep 10 2024
%e A143157 a(4) = 20 = sum of row 4 terms of triangle A143156, (7 + 6 + 4 + 3).
%e A143157 a(4) = 20 = partial sums of first 4 terms of A091512: (1 + 4 + 3 + 12).
%e A143157 a(4) = 20 = Sum_{j=1..4} j*A001511(j) = 1*1 + 2*2 + 3*1 + 4*3.
%t A143157 {0}~Join~Accumulate@ Array[IntegerExponent[(2 #)^#, 2] &, 56] (* _Michael De Vlieger_, Sep 29 2019 *)
%o A143157 (Python)
%o A143157 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
%Y A143157 Cf. A001511, A007814, A091512, A143156, A249152, A249153.
%K A143157 nonn,easy
%O A143157 0,3
%A A143157 _Gary W. Adamson_, Jul 27 2008
%E A143157 a(0) = 0 prepended and more terms computed by _Antti Karttunen_, Oct 25 2014