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%I #106 Jan 12 2024 06:08:30
%S 1,6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96,102,108,114,120,126,
%T 132,138,144,150,156,162,168,174,180,186,192,198,204,210,216,222,228,
%U 234,240,246,252,258,264,270,276,282,288,294,300,306,312,318,324,330,336,342,348
%N Coordination sequence for hexagonal lattice.
%C The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. It is also the planar net 3.3.3.3.3.3.
%C Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_6].
%C Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 20 ).
%C Also the Engel expansion of exp^(1/6); cf. A006784 for the Engel expansion definition. - _Benoit Cloitre_, Mar 03 2002
%C Numbers k such that k+floor(k/2) | k*floor(k/2). - _Wesley Ivan Hurt_, Dec 01 2020
%H T. D. Noe, <a href="/A008458/b008458.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Beck and S. Hosten, <a href="https://arxiv.org/abs/math/0508136">Cyclotomic polytopes and growth series of cyclotomic lattices</a>, arXiv:math/0508136 [math.CO], 2005-2006.
%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%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 4.1 p. 19.
%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 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/hxl">hxl</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 William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>
%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%H <a href="/index/Con#coordination_sequences">Index entries for coordination sequences</a>
%F G.f.: (1 + 4*x + x^2)/(1 - x)^2.
%F a(n) = A003215(n) - A003215(n-1), n > 0.
%F Equals binomial transform of [1, 5, 1, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Jul 08 2008
%F G.f.: Hypergeometric2F1([3,-2], [1], -x/(1-x)). - _Paul Barry_, Sep 18 2008
%F a(n) = 0^n + 6*n. - _Vincenzo Librandi_, Aug 21 2011
%F n*a(1) + (n-1)*a(2) + (n-2)*a(3) + ... + 2*a(n-1) + a(n) = n^3. - _Warren Breslow_, Oct 28 2013
%F E.g.f.: 1 + 6*x*exp(x). - _Stefano Spezia_, Jun 26 2022
%e From _Omar E. Pol_, Aug 20 2011: (Start)
%e Illustration of initial terms:
%e . o o o o o
%e . o o o o o o
%e . o o o o o o o
%e . o 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
%e . 1 o o o o o o o
%e . 6 o o o o o o
%e . 12 o o o o o
%e . 18
%e . 24
%e (End)
%e G.f. = 1 + 6*x + 12*x^2 + 18*x^3 + 24*x^4 + 30*x^5 + 36*x^6 + 42*x^7 + 48*x^8 + 54*x^9 + ...
%p 1, seq(6*n, n=1..65);
%t Join[{1},6*Range[60]] (* _Harvey P. Dale_, Jul 21 2013 *)
%t a[ n_] := Boole[n == 0] + 6 n; (* _Michael Somos_, May 21 2015 *)
%o (PARI) {a(n) = 6*n + (!n)};
%o (Magma) [0^n+6*n: n in [0..60] ]; // _Vincenzo Librandi_, Aug 21 2011
%o (Maxima) makelist(if n=0 then 1 else 6*n,n,0,65); /* _Martin Ettl_, Nov 12 2012 */
%o (SageMath) [6*n+int(n==0) for n in range(66)] # _G. C. Greubel_, May 25 2023
%Y Essentially the same as A008588.
%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. A032528. - _Omar E. Pol_, Aug 20 2011
%Y Cf. A048477 (binomial Transf.)
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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