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 A032765 #88 Aug 18 2025 06:55:12
%S A032765 0,1,2,5,8,11,16,21,26,33,40,47,56,65,74,85,96,107,120,133,146,161,
%T A032765 176,191,208,225,242,261,280,299,320,341,362,385,408,431,456,481,506,
%U A032765 533,560,587,616,645,674,705,736,767,800,833,866,901,936,971,1008
%N A032765 a(n) = floor(n*(n+1)*(n+2) / (n + n+1 + n+2)) = floor(n*(n+2)/3).
%C A032765 Satisfies a(n+1) - 2*a(n) + a(n-1) = (2/3)*(1 + w^(n+1) + w^(2*n+2)), a(0)=0 & a(1)=1 where w is the imaginary cubic root of unity. - _Robert G. Wilson v_, Jun 24 2002
%C A032765 First differences have this pattern: (+1) +1 +1 +3 +3 +3 +5 +5 +5 +7 +7 +7 +9 +9 +9. - _Alexandre Wajnberg_, Dec 19 2005
%C A032765 In Duistermaat (2010) section 11.3 The Planar Four-Bar Link on page 516: "It follows from (4.3.2) that the number of k-periodic fibers of the QRT automorphism, counted with multiplicities, is equal to nu((tau^S)^k) = 3*n^2 - 1 when k = 3*n, 3*n^2 + 2*n when k = 3*n + 1, 3*n^2 + 4*n + 1 when k = 3*n + 2, for every integer n." - _Michael Somos_, Mar 14 2023
%C A032765 a(n-1) is the maximum number of edges in a graph with vertices labeled from 1 to n, such that there is no triangle whose sum of vertex labels is divisible by n. - _Hoang Xuan Thanh_, Jun 11 2025
%D A032765 Johannes J. Duistermaat, Discrete Integrable Systems, Springer Science+Business Media, 2010.
%H A032765 Johannes Bader, <a href="https://www.sop.tik.ee.ethz.ch/people/baderj/other.html">Kobon Triangles</a>.
%H A032765 Gilles Clement and Johannes Bader, <a href="https://www.sop.tik.ee.ethz.ch/publicationListFiles/cb2007a.pdf">Tighter Upper Bound for the Number of Kobon Triangles</a>, Unpublished, 2007.
%H A032765 Gilles Clement and Johannes Bader, <a href="/A006066/a006066.pdf">Tighter Upper Bound for the Number of Kobon Triangles</a>, Unpublished, 2007. [Cached copy, with permission]
%H A032765 Taylor Short, <a href="https://arxiv.org/abs/1807.11355">The saturation number of carbon nanocones and nanotubes</a>, arXiv:1807.11355 [math.CO], 2018.
%H A032765 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KobonTriangle.html">Kobon Triangle</a>.
%H A032765 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A032765 From _Ralf Stephan_, May 05 2004: (Start)
%F A032765 a(n) = n^2 - ceiling(n*(n-1)/3).
%F A032765 G.f.: x*(1+2x^2-x^3)/((1+x+x^2)(1-x)^3). (End)
%F A032765 a(n) = floor(n*(n+2)/3). - Saburo Tamura, sent by _Alexandre Wajnberg_, Dec 19 2005
%F A032765 a(3*n - 1) = 3*n^2 - 1, a(3*n) = 3*n^2 + 2*n, a(3*n + 1) = 3*n^2 + 4*n + 1. - _Michael Somos_, Mar 14 2023
%F A032765 a(n+5) = A000217(n+5) - A001399(n+3) - A001399(n). - _Hoang Xuan Thanh_, Jun 11 2025
%F A032765 Sum_{n>=1} (-1)^(n+1)/a(n) = -1/4 + (Pi/(2*sqrt(3)))*(2-cosec(Pi/sqrt(3))). - _Amiram Eldar_, Jun 18 2025
%e A032765 G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 11*x^5 + 16*x^6 + 21*x^7 + 26*x^8 + ... - _Michael Somos_, Mar 14 2023
%e A032765 Let n = 5: a(5-1) = 8. Consider the graph G(5) with vertex set {1, 2, 3, 4, 5} and the edge set: E = {12, 23, 34, 45, 51, 13, 24, 35}, which contains 8 edges. The triangles in this graph are: {123}, {234}, {345}, {135}. Their vertex sums are: 1+2+3 = 6, 2+3+4 = 9, 3+4+5 = 12, 1+3+5=9. Since none of these sums are divisible by 5, the graph satisfies the required condition. If we add the edge 14, a new triangle {145} would appear, whose vertex sum is 1+4+5 = 10, divisible by 5. Similarly, if we add the edge 25, the triangle {235} would form, with vertex sum 2+3+5 = 10, also divisible by 5. - _Hoang Xuan Thanh_, Jun 11 2025
%p A032765 A032765:=n->floor(n*(n+2)/3); seq(A032765(n), n=0..100); # _Wesley Ivan Hurt_, Dec 20 2013
%t A032765 Table[Floor[n (n + 1)(n + 2)/(n + (n + 1) + (n + 2))], {n, 0, 55}]
%t A032765 Table[Floor[n (n + 2)/3], {n, 0, 100}] (* _Wesley Ivan Hurt_, Dec 20 2013 *)
%t A032765 LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 2, 5, 8}, 60] (* _Harvey P. Dale_, Jun 06 2016 *)
%t A032765 Table[(3 n (n + 2) + 2 Cos[2 n Pi/3] + 2 Sqrt[3] Sin[2 n Pi/3] - 2)/9, {n, 0, 20}] (* _Eric W. Weisstein_, Jul 27 2025 *)
%t A032765 CoefficientList[Series[x (-1 - 2 x^2 + x^3)/((-1 + x)^3 (1 + x + x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jul 27 2025 *)
%o A032765 (PARI) a(n)=n*(n+2)\3 \\ _Charles R Greathouse IV_, Jun 11 2015
%Y A032765 Cf. A001082, A032766, A000217, A001399.
%K A032765 nonn,easy
%O A032765 0,3
%A A032765 _Patrick De Geest_, May 15 1998
%E A032765 Name change suggested by _Wesley Ivan Hurt_, Dec 20 2013