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 A111927 #51 Feb 06 2025 12:43:48
%S A111927 0,0,0,1,4,10,21,42,84,169,340,682,1365,2730,5460,10921,21844,43690,
%T A111927 87381,174762,349524,699049,1398100,2796202,5592405,11184810,22369620,
%U A111927 44739241,89478484,178956970,357913941,715827882,1431655764,2863311529,5726623060
%N A111927 Expansion of x^3 / ((x-1)*(2*x-1)*(x^2-x+1)).
%C A111927 Binomial transform of sequence (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0). Note: the binomial transform of the sequence (0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A111926; the binomial transform of the sequence (0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A024495 (disregarding first two terms, which are both zero).
%C A111927 The sequence relates the calculation of the logarithm of the Twin Prime Constants of order 3 to the sequence of prime zeta functions, see definition 7 in arXiv:0903.2514. - _R. J. Mathar_, Mar 28 2009
%H A111927 Colin Barker, <a href="/A111927/b111927.txt">Table of n, a(n) for n = 0..1000</a>
%H A111927 Antoine-Augustin Cournot, <a href="https://archive.org/download/bulletindesscie22unkngoog/">Solution d'un problème d'analyse combinatoire</a>, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at <a href="http://books.google.com.au/books?id=B-v-eXuvoG4C">Google Books</a>. Page 97 case p=3 formula y^(0) = a(n). (But misprint "- (2/3)*cos" should be "+ (2/3)*cos".)
%H A111927 R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, arXiv:0903.2514 [math.NT], 2009-2011.
%H A111927 Christian Ramus, <a href="http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN243919689_0011">Solution générale d'un problème d'analyse combinatoire</a>, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(0) = a(n). (But misprint "+ (1/3)*cos" should be "+ (2/3)*cos", per the general case equation A page 354.)
%H A111927 Kevin Ryde, <a href="http://user42.tuxfamily.org/terdragon/index.html">Iterations of the Terdragon Curve</a>, section Lines, quantity Lines_k(2) = a(k+1).
%H A111927 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-2).
%F A111927 a(n+2) - a(n+1) + a(n) = A000225(n).
%F A111927 a(n) - a(n-1) = A024495(n-1).
%F A111927 From _Colin Barker_, Feb 10 2017: (Start)
%F A111927 a(n) = 2^n/3 + 2*cos((Pi*n)/3)/3 - 1. [Cournot]
%F A111927 a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n > 3. (End)
%F A111927 a(n) = (2^n+A087204(n))/3 - 1. - _R. J. Mathar_, Aug 07 2017
%F A111927 a(n) = (1/3)*Sum_{k=0..n-1} binomial(n, 3*floor(k/3)+3). - _Taras Goy_, Jan 26 2025
%F A111927 E.g.f.: (exp(x)*(exp(x) - 3) + 2*exp(x/2)*cos(sqrt(3)*x/2))/3. - _Stefano Spezia_, Feb 06 2025
%p A111927 seq(sum(binomial(n, k*3), k=1..n), n=0..33); # _Zerinvary Lajos_, Oct 23 2007
%t A111927 LinearRecurrence[{4,-6,5,-2},{0,0,0,1},40] (* _Harvey P. Dale_, Jul 04 2017 *)
%o A111927 (PARI) concat(vector(3), Vec(x^3/((x-1)*(2*x-1)*(x^2-x+1)) + O(x^40))) \\ _Colin Barker_, Feb 10 2017
%Y A111927 Cf. A000295, A111926, A024495, A131708.
%K A111927 easy,nonn
%O A111927 0,5
%A A111927 _Creighton Dement_, Aug 21 2005