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 A076178 #30 Nov 13 2024 15:19:23 %S A076178 0,0,0,2,2,4,6,10,10,12,14,18,20,24,28,34,34,36,38,42,44,48,52,58,60, %T A076178 64,68,74,78,84,90,98,98,100,102,106,108,112,116,122,124,128,132,138, %U A076178 142,148,154,162,164,168,172,178,182,188,194,202,206,212,218,226,232 %N A076178 a(n) = 2*n^2 - A077071(n). %H A076178 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf">Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications</a>, preprint, 2016. %H A076178 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="http://doi.org/10.1145/3127585">Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications</a>, ACM Transactions on Algorithms 13:4 (2017), #47. %H A076178 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences with (relatively) simple generating functions</a>, 2004. %H A076178 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions (ps file)</a>. %H A076178 Ralf Stephan, <a href="/A067699/a067699.pdf">Table of generating functions (pdf file)</a>. %F A076178 a(n) = A001105(n) - A077071(n). - _Omar E. Pol_, Nov 13 2024 %o A076178 (PARI) a(n)=2*n^2-sum(k=0,n,-valuation(polcoeff(pollegendre(2*n),2*k),2)) %o A076178 (Python) %o A076178 def A076178(n): return ((n+1)*n.bit_count()-n<<1)+sum((m:=1<<j)*((k:=n>>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1)) # _Chai Wah Wu_, Nov 12 2024 %Y A076178 Equals 2 * A078903(n). %Y A076178 Cf. A001105, A067699, A077070, A077071. %K A076178 nonn %O A076178 0,4 %A A076178 _Benoit Cloitre_, Nov 01 2002