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 A028896 #155 Aug 07 2025 14:47:03
%S A028896 0,6,18,36,60,90,126,168,216,270,330,396,468,546,630,720,816,918,1026,
%T A028896 1140,1260,1386,1518,1656,1800,1950,2106,2268,2436,2610,2790,2976,
%U A028896 3168,3366,3570,3780,3996,4218,4446,4680,4920,5166,5418,5676
%N A028896 6 times triangular numbers: a(n) = 3*n*(n+1).
%C A028896 From _Floor van Lamoen_, Jul 21 2001: (Start)
%C A028896 Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0, 6, ...
%C A028896 The spiral begins:
%C A028896 85--84--83--82--81--80
%C A028896 / \
%C A028896 86 56--55--54--53--52 79
%C A028896 / / \ \
%C A028896 87 57 33--32--31--30 51 78
%C A028896 / / / \ \ \
%C A028896 88 58 34 16--15--14 29 50 77
%C A028896 / / / / \ \ \ \
%C A028896 89 59 35 17 5---4 13 28 49 76
%C A028896 / / / / / \ \ \ \ \
%C A028896 <==90==60==36==18===6===0 3 12 27 48 75
%C A028896 / / / / / / / / / /
%C A028896 61 37 19 7 1---2 11 26 47 74
%C A028896 \ \ \ \ / / / /
%C A028896 62 38 20 8---9--10 25 46 73
%C A028896 \ \ \ / / /
%C A028896 63 39 21--22--23--24 45 72
%C A028896 \ \ / /
%C A028896 64 40--41--42--43--44 71
%C A028896 \ /
%C A028896 65--66--67--68--69--70
%C A028896 (End)
%C A028896 If Y is a 4-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-4)-subsets of X having exactly two elements in common with Y. - _Milan Janjic_, Dec 28 2007
%C A028896 a(n) is the maximal number of points of intersection of n+1 distinct triangles drawn in the plane. For example, two triangles can intersect in at most a(1) = 6 points (as illustrated in the Star of David configuration). - Terry Stickels (Terrystickels(AT)aol.com), Jul 12 2008
%C A028896 Also sequence found by reading the line from 0, in the direction 0, 6, ... and the same line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Axis perpendicular to A195143 in the same spiral. - _Omar E. Pol_, Sep 18 2011
%C A028896 Partial sums of A008588. - _R. J. Mathar_, Aug 28 2014
%C A028896 Also the number of 5-cycles in the (n+5)-triangular honeycomb acute knight graph. - _Eric W. Weisstein_, Jul 27 2017
%C A028896 a(n-4) is the maximum irregularity over all maximal 3-degenerate graphs with n vertices. The extremal graphs are 3-stars (K_3 joined to n-3 independent vertices). (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - _Allan Bickle_, May 29 2023
%H A028896 Ivan Panchenko, <a href="/A028896/b028896.txt">Table of n, a(n) for n = 0..1000</a>
%H A028896 Allan Bickle and Zhongyuan Che, <a href="https://doi.org/10.1016/j.dam.2023.01.020">Irregularities of Maximal k-degenerate Graphs</a>, Discrete Applied Math. 331 (2023) 70-87.
%H A028896 Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H A028896 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A028896 Enrique Navarrete and Daniel Orellana, <a href="https://arxiv.org/abs/1907.10023">Finding Prime Numbers as Fixed Points of Sequences</a>, arXiv:1907.10023 [math.NT], 2019.
%H A028896 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H A028896 Leo Tavares, <a href="/A028896/a028896.jpg">Illustration: Centroid Hexagons</a>.
%H A028896 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H A028896 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A028896 O.g.f.: 6*x/(1 - x)^3.
%F A028896 E.g.f.: 3*x*(x + 2)*exp(x). - _G. C. Greubel_, Aug 19 2017
%F A028896 a(n) = 6*A000217(n).
%F A028896 a(n) = polygorial(3, n+1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003
%F A028896 From _Zerinvary Lajos_, Mar 06 2007: (Start)
%F A028896 a(n) = A049598(n)/2.
%F A028896 a(n) = A124080(n) - A046092(n).
%F A028896 a(n) = A033996(n) - A002378(n). (End)
%F A028896 a(n) = A002378(n)*3 = A045943(n)*2. - _Omar E. Pol_, Dec 12 2008
%F A028896 a(n) = a(n-1) + 6*n for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 05 2010
%F A028896 a(n) = A003215(n) - 1. - _Omar E. Pol_, Oct 03 2011
%F A028896 From _Philippe Deléham_, Mar 26 2013: (Start)
%F A028896 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=6, a(2)=18.
%F A028896 a(n) = A174709(6*n + 5). (End)
%F A028896 a(n) = A049450(n) + 4*n. - _Lear Young_, Apr 24 2014
%F A028896 a(n) = Sum_{i = n..2*n} 2*i. - _Bruno Berselli_, Feb 14 2018
%F A028896 a(n) = A320047(1, n, 1). - _Kolosov Petro_, Oct 04 2018
%F A028896 a(n) = T(3*n) - T(2*n-2) + T(n-2), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - _Charlie Marion_, Dec 04 2020
%F A028896 From _Amiram Eldar_, Feb 15 2022: (Start)
%F A028896 Sum_{n>=1} 1/a(n) = 1/3.
%F A028896 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 1/3. (End)
%F A028896 From _Amiram Eldar_, Feb 21 2023: (Start)
%F A028896 Product_{n>=1} (1 - 1/a(n)) = -(3/Pi)*cos(sqrt(7/3)*Pi/2).
%F A028896 Product_{n>=1} (1 + 1/a(n)) = (3/Pi)*cosh(Pi/(2*sqrt(3))). (End)
%p A028896 [seq(6*binomial(n,2),n=1..44)]; # _Zerinvary Lajos_, Nov 24 2006
%t A028896 6 Accumulate[Range[0, 50]] (* _Harvey P. Dale_, Mar 05 2012 *)
%t A028896 6 PolygonalNumber[Range[0, 20]] (* _Eric W. Weisstein_, Jul 27 2017 *)
%t A028896 LinearRecurrence[{3, -3, 1}, {0, 6, 18}, 20] (* _Eric W. Weisstein_, Jul 27 2017 *)
%o A028896 (Magma) [3*n*(n+1): n in [0..50]]; // _Wesley Ivan Hurt_, Jun 09 2014
%o A028896 (PARI) a(n)=3*n*(n+1) \\ _Charles R Greathouse IV_, Sep 24 2015
%o A028896 (PARI) first(n) = Vec(6*x/(1 - x)^3 + O(x^n), -n) \\ _Iain Fox_, Feb 14 2018
%o A028896 (GAP) List([0..44],n->3*n*(n+1)); # _Muniru A Asiru_, Mar 15 2019
%o A028896 (Python)
%o A028896 def A028896(n): return 3*n*(n+1) # _Chai Wah Wu_, Aug 07 2025
%Y A028896 Cf. A000217, A000567, A003215, A008588, A024966, A028895, A033996, A046092, A049598, A084939, A084940, A084941, A084942, A084943, A084944, A124080.
%Y A028896 Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A152773 (6-cycles).
%Y A028896 Cf. A007531.
%Y A028896 The partial sums give A007531. - _Leo Tavares_, Jan 22 2022
%Y A028896 Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).
%K A028896 nonn,easy
%O A028896 0,2
%A A028896 Joe Keane (jgk(AT)jgk.org), Dec 11 1999