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%I #169 Oct 26 2023 08:46:28
%S 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 72,75,78,81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,
%U 129,132,135,138,141,144,147,150,153,156,159,162,165,168,171,174,177,180,183,186
%N Expansion of (1 + x + x^2)/(1 - x)^2.
%C Also the Engel expansion of exp^(1/3); cf. A006784 for the Engel expansion definition. - _Benoit Cloitre_, Mar 03 2002
%C 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 Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_3].
%C 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 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 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 a(n) = partial sums of A158799(n). Partial sums of a(n) = A005448(n). - _Jaroslav Krizek_, Dec 06 2009
%C 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 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 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 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 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 J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.
%H Vincenzo Librandi, <a href="/A008486/b008486.txt">Table of n, a(n) for n = 0..10000</a>
%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H 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 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 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 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 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H 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 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 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 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H Kival Ngaokrajang, <a href="/A008486/a008486_2.pdf">Illustration of polygon expansions (planar net)</a>
%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/hcb">hcb</a>
%H 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 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 University of Manchester, <a href="http://www.graphene.manchester.ac.uk/">Graphene</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Thomson_problem">Thomson problem</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(0) = 1; a(n) = 3*n = A008585(n), n >= 1.
%F Euler transform of length 3 sequence [3, 0, -1]. - _Michael Somos_, Aug 04 2009
%F a(n) = a(n-1) + 3 for n >= 2. - _Jaroslav Krizek_, Nov 18 2009
%F a(n) = 0^n + 3*n. - _Vincenzo Librandi_, Aug 21 2011
%F a(n) = -a(-n) unless n = 0. - _Michael Somos_, May 05 2015
%F E.g.f.: 1 + 3*exp(x)*x. - _Stefano Spezia_, Aug 07 2022
%e 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 From _Omar E. Pol_, Aug 20 2011: (Start)
%e Illustration of initial terms as triangles:
%e . o
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%e . o o o o o o o
%e . o o o o o o o o o
%e . o o o o o o o o o o o o o o o o o o o o o
%e .
%e . 1 3 6 9 12 15
%e (End)
%t CoefficientList[Series[(1 + x + x^2) / (1 - x)^2, {x, 0, 80}], x] (* _Vincenzo Librandi_, Nov 23 2014 *)
%t a[ n_] := If[ n == 0, 1, 3 n]; (* _Michael Somos_, Apr 17 2015 *)
%o (PARI) {a(n) = if( n==0, 1, 3 * n)}; /* _Michael Somos_, May 05 2015 */
%o (Magma) [0^n+3*n: n in [0..90] ]; // _Vincenzo Librandi_, Aug 21 2011
%o (Haskell)
%o a008486 0 = 1; a008486 n = 3 * n
%o a008486_list = 1 : [3, 6 ..] -- _Reinhard Zumkeller_, Apr 17 2015
%Y Partial sums give A005448.
%Y 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 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 Cf. A000217, A001651, A005448, A006784, A008585, A022424, A113062, A128834, A139250, A158799.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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