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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 (list; graph; refs; listen; history; text; internal format)
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 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 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 (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], 2005-2006.

Branko Gr├╝nbaum 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. [alternate link]

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

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

S. Kitaev, On multi-avoidance 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 A-numbers [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)/ (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

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

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Last modified October 23 06:55 EDT 2017. Contains 293783 sequences.