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A008574
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Expansion of (1+x)^2 / (1-x)^2 (coordination sequence for square lattice).
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30
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
Number of squares in an n X n board with all non-perimeter squares removed. - Jon Perry, Jul 27 2003
Jon Perry's comment considers this sequence with a different offset, namely one such that a(2) = 4 rather than 8. For boards bigger than 2 x 2, the formula n^2 - ((n - 2)^2) = 4(n - 1) can be used, meaning that we remove the biggest overall square that is still smaller than the whole square and then count the remaining unit squares; e.g., from a 5 x 5 board we remove a 3 x 3 board, leaving 16 unit squares. 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
Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_4].
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REFERENCES
| D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
Index entries for sequences related to linear recurrences with constant coefficients
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv math.CO/0508136.
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).
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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
Row sums of triangle A130323: (1; 3,1; 5,2,1; 7,3,1,1;...). - Gary W. Adamson, May 24 2007
Row sums of triangle A131032: (1; 3,1; 5,2,1; 7,2,2,1;...). - Gary W. Adamson, Jun 10 2007
G.f.: exp(4*atanh(x)) [From Jaume Oliver Lafont, Oct 20 2009]
a(n)=a(n-1)+4, n>1. [From Vincenzo Librandi, Dec 31 2010]
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EXAMPLE
| Contribution from Omar E. Pol, Aug 20 2011 (Start):
Illustration of initial terms as squares:
. 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)
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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]] (* From Harvey P. Dale, Aug 19 2011 *)
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PROG
| (PARI) {a(n)= 4*n+!n} /* Michael Somos Apr 16 2007 */
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CROSSREFS
| Cf. A054275, A054410, A054389, A054764.
Convolution square of A040000.
Cf. A130323.
Cf. A131032.
Sequence in context: A161352 A008586 A059558 * A189917 A172326 A085127
Adjacent sequences: A008571 A008572 A008573 * A008575 A008576 A008577
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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