

A008574


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


61



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
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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+106 = 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 period3 structure continues for ever, 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 nth 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 12gons. (End)
Also the coordination sequence for a 6.6.6.6 point in the 3transitive 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 2dimensional cyclotomic lattice Z[zeta_4].
Susceptibility series H_1 for 2dimensional 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 (kitaev(AT)ms.uky.edu), Nov 11 2004
Central terms of the triangle in A118013.  Reinhard Zumkeller, Apr 10 2006


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000
Joerg Arndt, The 3.4.6.4 net
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 20052006.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227247.
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167189. [alternate link]
D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 6980.
S. Kitaev, On multiavoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
S. Kitaev, On multiavoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
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 Anumbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (2,1).


FORMULA

Binomial transform is A000337.  Paul Barry, Jul 21 2003
Euler transform of length 2 sequence [ 4, 2].  Michael Somos, Apr 16 2007
G.f.: ((1+x)/ (1x))^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(n1)+4, n>1.  Vincenzo Librandi, Dec 31 2010
a(n) = A005408(n1)+A005408(n), n>1.  Ivan N. Ianakiev, Jul 16 2012


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 *)


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. 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).
See also A265045, A265046.
Sequence in context: A161352 A008586 A059558 * A189917 A172326 A085127
Adjacent sequences: A008571 A008572 A008573 * A008575 A008576 A008577


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane. Entry revised Aug 24 2014.


STATUS

approved



