A008574 a(0) = 1, thereafter a(n) = 4n.

1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset

0,2

Comments

Number of squares on the perimeter of an (n+1) X (n+1) board. - Jon Perry, Jul 27 2003

Coordination sequence for square lattice (or equivalently the planar net 4.4.4.4).

Apparently also the coordination sequence for the planar net 3.4.6.4. - Darrah Chavey, Nov 23 2014

From N. J. A. Sloane, Nov 26 2014: (Start)

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).

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.

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).

It is clear from the net that this period-3 structure continues forever, and establishes the theorem.

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.

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)

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

Also the coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_4].

Susceptibility series H_1 for 2-dimensional Ising model (divided by 2).

Also the Engel expansion of exp^(1/4); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002

This sequence differs from A008586, multiples of 4, only in its initial term. - Alonso del Arte, Apr 14 2011

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

Central terms of the triangle in A118013. - Reinhard Zumkeller, Apr 10 2006

Also the coordination sequence for the htb net. - N. J. A. Sloane, Mar 31 2018

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

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Joerg Arndt, The 3.4.6.4 net

Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.

Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.

Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 7.

Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers

Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.

Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.2 p. 20.

Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.

A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.

D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.

Tom Karzes, Tiling Coordination Sequences

Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.

Reticular Chemistry Structure Resource, sql and htb

Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.

N. J. A. Sloane, Illustration of points in shells 0 through 12 of the 3.4.6.4 planar net (see Comments for discussion)

N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]

N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]

N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Michael Somos, Rational Function Multiplicative Coefficients

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

Binomial transform is A000337 (dropping the 0 there). - Paul Barry, Jul 21 2003

Euler transform of length 2 sequence [4, -2]. - Michael Somos, Apr 16 2007

G.f.: ((1 + x) / (1 - x))^2. E.g.f.: 1 + 4*x*exp(x). - Michael Somos, Apr 16 2007

a(-n) = -a(n) unless n = 0. - Michael Somos, Apr 16 2007

G.f.: exp(4*atanh(x)). - Jaume Oliver Lafont, Oct 20 2009

a(n) = a(n-1) + 4, n > 1. - Vincenzo Librandi, Dec 31 2010

a(n) = A005408(n-1) + A005408(n), n > 1. - Ivan N. Ianakiev, Jul 16 2012

a(n) = 4*n = A008586(n), n >= 1. - Tom Karzes, Apr 01 2020

Example

From Omar E. Pol, Aug 20 2011 (Start):
Illustration of initial terms as perimeters of squares (cf. Perry's comment above):
.                                         o o o o o o
.                             o o o o o   o         o
.                   o o o o   o       o   o         o
.           o o o   o     o   o       o   o         o
.     o o   o   o   o     o   o       o   o         o
. o   o o   o o o   o o o o   o o o o o   o o o o o o
.
. 1    4      8        12         16           20
(End)

Mathematica

f[0] = 1; f[n_] := 4 n; Array[f, 59, 0] (* or *)
CoefficientList[ Series[(1 + x)^2/(1 - x)^2, {x, 0, 58}], x] (* Robert G. Wilson v, Jan 02 2011 *)
Join[{1}, Range[4, 232, 4]] (* Harvey P. Dale, Aug 19 2011 *)
a[ n_] := 4 n + Boole[n == 0]; (* Michael Somos, Jan 07 2019 *)

Prog

(PARI) {a(n) = 4*n + !n}; /* Michael Somos, Apr 16 2007 */
(Haskell)
a008574 0 = 1; a008574 n = 4 * n
a008574_list = 1 : [4, 8 ..]  -- Reinhard Zumkeller, Apr 16 2015

Crossrefs

Cf. A001844 (partial sums), A008586, A054275, A054410, A054389, A054764.

Convolution square of A040000.

Row sums of A130323 and A131032.

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).

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.

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.

See also A265045, A265046.

Sequence in context: A295774 A008586 A059558 * A189917 A172326 A329975

Adjacent sequences: A008571 A008572 A008573 * A008575 A008576 A008577

Keyword

nonn,nice,easy

Author

N. J. A. Sloane; entry revised Aug 24 2014

Status

approved