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 A008486 #169 Oct 26 2023 08:46:28
%S A008486 1,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,
%T A008486 72,75,78,81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,
%U A008486 129,132,135,138,141,144,147,150,153,156,159,162,165,168,171,174,177,180,183,186
%N A008486 Expansion of (1 + x + x^2)/(1 - x)^2.
%C A008486 Also the Engel expansion of exp^(1/3); cf. A006784 for the Engel expansion definition. - _Benoit Cloitre_, Mar 03 2002
%C A008486 Coordination sequence for planar net 6^3 (the graphite net, or the graphene crystal) - that is, the number of atoms at graph distance n from any fixed atom. Also for the hcb or honeycomb net. - _N. J. A. Sloane_, Jan 06 2013, Mar 31 2018
%C A008486 Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_3].
%C A008486 Conjecture: This is also the maximum number of edges possible in a planar simple graph with n+2 vertices. - _Dmitry Kamenetsky_, Jun 29 2008
%C A008486 The conjecture is correct. Proof: For n=0 the theorem holds, the maximum planar graph has n+2=2 vertices and 1 edge. Now suppose that we have a connected planar graph with at least 3 vertices. If it contains a face that is not a triangle, we can add an edge that divides this face into two without breaking its planarity. Hence all maximum planar graphs are triangulations. Euler's formula for planar graphs states that in any planar simple graph with V vertices, E edges and F faces we have V+F-E=2. If all faces are triangles, then F=2E/3, which gives us E=3V-6. Hence for n>0 each maximum planar simple graph with n+2 vertices has 3n edges. - _Michal Forisek_, Apr 23 2009
%C A008486 a(n) = sum of natural numbers m such that n - 1 <= m <= n + 1. Generalization: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). - _Jaroslav Krizek_, Nov 18 2009
%C A008486 a(n) = partial sums of A158799(n). Partial sums of a(n) = A005448(n). - _Jaroslav Krizek_, Dec 06 2009
%C A008486 Integers n dividing a(n) = a(n-1) - a(n-2) with initial conditions a(0)=0, a(1)=1 (see A128834 with offset 0). - _Thomas M. Bridge_, Nov 03 2013
%C A008486 a(n) is conjectured to be the number of polygons added after n iterations of the polygon expansions (type A, B, C, D & E) shown in the Ngaokrajang link. The patterns are supposed to become the planar Archimedean net 3.3.3.3.3.3, 3.6.3.6, 3.12.12, 3.3.3.3.6 and 4.6.12 respectively when n - > infinity. - _Kival Ngaokrajang_, Dec 28 2014
%C A008486 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I. - _Ray Chandler_, Nov 21 2016
%C A008486 Conjecture: let m = n + 2, p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p (see the Wikipedia link below), and f(m) = a(n) - q. Then f(m) would be the solution of the Thompson problem for all m in 3-space. - _Sergey Pavlov_, Feb 03 2017
%C A008486 Also, sequence defined by a(0)=1, a(1)=3, c(0)=2, c(1)=4; and thereafter a(n) = c(n-1) + c(n-2), and c consists of the numbers missing from a (see A001651). - _Ivan Neretin_, Mar 28 2017
%D A008486 J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.
%H A008486 Vincenzo Librandi, <a href="/A008486/b008486.txt">Table of n, a(n) for n = 0..10000</a>
%H A008486 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A008486 M. Beck and S. Hosten, <a href="http://arxiv.org/abs/math/0508136">Cyclotomic polytopes and growth series of cyclotomic lattices</a>, arXiv:math/0508136 [math.CO], 2005-2006.
%H A008486 Jean-Guillaume Eon, <a href="http://dx.doi.org/10.1107/S0108767301016609">Algebraic determination of generating functions for coordination sequences in crystal structures</a>, Acta Cryst. A58 (2002), 47-53.
%H A008486 Jean-Guillaume Eon, <a href="https://doi.org/10.3390/sym10020035">Symmetry and Topology: The 11 Uninodal Planar Nets Revisited</a>, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 5.
%H A008486 A. S. Fraenkel, <a href="http://www.emis.de/journals/INTEGERS/papers/eg6/eg6.Abstract.html">New games related to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)
%H A008486 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A008486 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">on arXiv</a>, arXiv:1803.08530 [math.CO], 2018-2019.
%H A008486 Rostislav Grigorchuk and Cosmas Kravaris, <a href="https://arxiv.org/abs/2012.13661">On the growth of the wallpaper groups</a>, arXiv:2012.13661 [math.GR], 2020. See section 4.3 p. 20.
%H A008486 Branko Grünbaum and Geoffrey C. Shephard, <a href="http://www.jstor.org/stable/2689529">Tilings by regular polygons</a>, Mathematics Magazine, 50 (1977), 227-247.
%H A008486 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H A008486 Kival Ngaokrajang, <a href="/A008486/a008486_2.pdf">Illustration of polygon expansions (planar net)</a>
%H A008486 Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/hcb">hcb</a>
%H A008486 N. J. A. Sloane, <a href="/A008576/a008576.png">The uniform planar nets and their A-numbers</a> [Annotated scanned figure from Gruenbaum and Shephard (1977)]
%H A008486 N. J. A. Sloane, <a href="/A296368/a296368_2.png">Overview of coordination sequences of Laves tilings</a> [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
%H A008486 University of Manchester, <a href="http://www.graphene.manchester.ac.uk/">Graphene</a>
%H A008486 Wikipedia, <a href="https://en.wikipedia.org/wiki/Thomson_problem">Thomson problem</a>
%H A008486 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A008486 a(0) = 1; a(n) = 3*n = A008585(n), n >= 1.
%F A008486 Euler transform of length 3 sequence [3, 0, -1]. - _Michael Somos_, Aug 04 2009
%F A008486 a(n) = a(n-1) + 3 for n >= 2. - _Jaroslav Krizek_, Nov 18 2009
%F A008486 a(n) = 0^n + 3*n. - _Vincenzo Librandi_, Aug 21 2011
%F A008486 a(n) = -a(-n) unless n = 0. - _Michael Somos_, May 05 2015
%F A008486 E.g.f.: 1 + 3*exp(x)*x. - _Stefano Spezia_, Aug 07 2022
%e A008486 G.f. = 1 + 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + 24*x^8 + ...
%e A008486 From _Omar E. Pol_, Aug 20 2011: (Start)
%e A008486 Illustration of initial terms as triangles:
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%e A008486 .
%e A008486 . 1 3 6 9 12 15
%e A008486 (End)
%t A008486 CoefficientList[Series[(1 + x + x^2) / (1 - x)^2, {x, 0, 80}], x] (* _Vincenzo Librandi_, Nov 23 2014 *)
%t A008486 a[ n_] := If[ n == 0, 1, 3 n]; (* _Michael Somos_, Apr 17 2015 *)
%o A008486 (PARI) {a(n) = if( n==0, 1, 3 * n)}; /* _Michael Somos_, May 05 2015 */
%o A008486 (Magma) [0^n+3*n: n in [0..90] ]; // _Vincenzo Librandi_, Aug 21 2011
%o A008486 (Haskell)
%o A008486 a008486 0 = 1; a008486 n = 3 * n
%o A008486 a008486_list = 1 : [3, 6 ..] -- _Reinhard Zumkeller_, Apr 17 2015
%Y A008486 Partial sums give A005448.
%Y A008486 List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574(4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579(3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529(3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
%Y A008486 List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
%Y A008486 Cf. A000217, A001651, A005448, A006784, A008585, A022424, A113062, A128834, A139250, A158799.
%K A008486 nonn,easy
%O A008486 0,2
%A A008486 _N. J. A. Sloane_