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%I #183 Jan 12 2024 06:22:00
%S 1,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,
%T 96,100,104,108,112,116,120,124,128,132,136,140,144,148,152,156,160,
%U 164,168,172,176,180,184,188,192,196,200,204,208,212,216,220,224,228,232
%N a(0) = 1, thereafter a(n) = 4n.
%C Number of squares on the perimeter of an (n+1) X (n+1) board. - _Jon Perry_, Jul 27 2003
%C Coordination sequence for square lattice (or equivalently the planar net 4.4.4.4).
%C Apparently also the coordination sequence for the planar net 3.4.6.4. - _Darrah Chavey_, Nov 23 2014
%C From _N. J. A. Sloane_, Nov 26 2014: (Start)
%C I confirm that this is indeed the coordination sequence for the planar net 3.4.6.4. The points at graph distance n from a fixed point in this net essentially lie on a hexagon (see illustration in link).
%C If n = 3k, k >= 1, there are 2k + 1 nodes on each edge of the hexagon. This counts the corners of the hexagon twice, so the number of points in the shell is 6(2k + 1) - 6 = 4n. If n = 3k + 1, the numbers of points on the six edges of the hexagon are 2k + 2 (4 times) and 2k + 1 (twice), for a total of 12k + 10 - 6 = 4n. If n = 3k + 2 the numbers are 2k + 2 (4 times) and 2k + 3 twice, and again we get 4n points.
%C The illustration shows shells 0 through 12, as well as the hexagons formed by shells 9 (green, 36 points), 10 (black, 40 points), 11 (red, 44 points), and 12 (blue, 48 points).
%C It is clear from the net that this period-3 structure continues forever, and establishes the theorem.
%C In contrast, for the 4.4.4.4 planar net, the successive shells are diamonds instead of hexagons, and again the n-th shell (n > 0) contains 4n points.
%C Of course the two nets are very different, since 4.4.4.4 has the symmetry of the square, while 3.4.6.4 has only mirror symmetry (with respect to a point), and has the symmetry of a regular hexagon with respect to the center of any of the 12-gons. (End)
%C Also the coordination sequence for a 6.6.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6}, see A265045, A265046. - _N. J. A. Sloane_, Dec 27 2015
%C Also the coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_4].
%C Susceptibility series H_1 for 2-dimensional Ising model (divided by 2).
%C Also the Engel expansion of exp^(1/4); cf. A006784 for the Engel expansion definition. - _Benoit Cloitre_, Mar 03 2002
%C This sequence differs from A008586, multiples of 4, only in its initial term. - _Alonso del Arte_, Apr 14 2011
%C Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00,0), (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2 and j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - _Sergey Kitaev_, Nov 11 2004
%C Central terms of the triangle in A118013. - _Reinhard Zumkeller_, Apr 10 2006
%C Also the coordination sequence for the htb net. - _N. J. A. Sloane_, Mar 31 2018
%C This is almost certainly also the coordination sequence for Dual(3.3.4.3.4) with respect to a tetravalent node. - _Tom Karzes_, Apr 01 2020
%C Minimal number of segments (equivalently, corners) in a rook circuit of a 2n X 2n board (maximal number is A085622). - _Ruediger Jehn_, Jan 02 2021
%H T. D. Noe, <a href="/A008574/b008574.txt">Table of n, a(n) for n = 0..1000</a>
%H Joerg Arndt, <a href="/A008574/a008574.pdf">The 3.4.6.4 net</a>
%H Matthias Beck and Serkan 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 Pierre de la Harpe, <a href="https://arxiv.org/abs/2106.02499">On the prehistory of growth of groups</a>, arXiv:2106.02499 [math.GR], 2021.
%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 7.
%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 4.2 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 A. J. Guttmann, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00262-9">Indicators of solvability for lattice models</a>, Discrete Math., 217 (2000), 167-189.
%H D. Hansel et al., <a href="http://dx.doi.org/10.1007/BF01010400">Analytical properties of the anisotropic cubic Ising model</a>, J. Stat. Phys., 48 (1987), 69-80.
%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H Sergey Kitaev, <a href="http://www.emis.de/journals/INTEGERS/papers/e21/e21.Abstract.html">On multi-avoidance of right angled numbered polyomino patterns</a>, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/sql">sql</a> and <a href="http://rcsr.net/layers/htb">htb</a>
%H Anton Shutov and Andrey Maleev, <a href="https://doi.org/10.1515/zkri-2020-0002">Coordination sequences of 2-uniform graphs</a>, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
%H N. J. A. Sloane, <a href="/A008574/a008574.png">Illustration of points in shells 0 through 12 of the 3.4.6.4 planar net</a> (see Comments for discussion)
%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 N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides.</a> (Mentions this sequence)
%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.html">Rational Function Multiplicative Coefficients</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F Binomial transform is A000337 (dropping the 0 there). - _Paul Barry_, Jul 21 2003
%F Euler transform of length 2 sequence [4, -2]. - _Michael Somos_, Apr 16 2007
%F G.f.: ((1 + x) / (1 - x))^2. E.g.f.: 1 + 4*x*exp(x). - _Michael Somos_, Apr 16 2007
%F a(-n) = -a(n) unless n = 0. - _Michael Somos_, Apr 16 2007
%F G.f.: exp(4*atanh(x)). - _Jaume Oliver Lafont_, Oct 20 2009
%F a(n) = a(n-1) + 4, n > 1. - _Vincenzo Librandi_, Dec 31 2010
%F a(n) = A005408(n-1) + A005408(n), n > 1. - _Ivan N. Ianakiev_, Jul 16 2012
%F a(n) = 4*n = A008586(n), n >= 1. - _Tom Karzes_, Apr 01 2020
%e From _Omar E. Pol_, Aug 20 2011 (Start):
%e Illustration of initial terms as perimeters of squares (cf. Perry's comment above):
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%e .
%e . 1 4 8 12 16 20
%e (End)
%t f[0] = 1; f[n_] := 4 n; Array[f, 59, 0] (* or *)
%t CoefficientList[ Series[(1 + x)^2/(1 - x)^2, {x, 0, 58}], x] (* _Robert G. Wilson v_, Jan 02 2011 *)
%t Join[{1},Range[4,232,4]] (* _Harvey P. Dale_, Aug 19 2011 *)
%t a[ n_] := 4 n + Boole[n == 0]; (* _Michael Somos_, Jan 07 2019 *)
%o (PARI) {a(n) = 4*n + !n}; /* _Michael Somos_, Apr 16 2007 */
%o (Haskell)
%o a008574 0 = 1; a008574 n = 4 * n
%o a008574_list = 1 : [4, 8 ..] -- _Reinhard Zumkeller_, Apr 16 2015
%Y Cf. A001844 (partial sums), A008586, A054275, A054410, A054389, A054764.
%Y Convolution square of A040000.
%Y Row sums of A130323 and A131032.
%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 Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
%Y See also A265045, A265046.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_; entry revised Aug 24 2014
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