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

Author

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

Keywords

• nonn
• nice
• easy

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

Examples

			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)
		

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).

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.

Programs

Haskell

a008574 0 = 1; a008574 n = 4 * n
a008574_list = 1 : [4, 8 ..]  -- Reinhard Zumkeller, Apr 16 2015

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 *)

PARI

{a(n) = 4*n + !n}; /* Michael Somos, Apr 16 2007 */

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