%I A008574
%S A008574 1,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,
%T A008574 96,100,104,108,112,116,120,124,128,132,136,140,144,148,152,156,160,164,
%U A008574 168,172,176,180,184,188,192,196,200,204,208,212,216,220,224,228,232
%N A008574 Expansion of (1+x)^2 / (1-x)^2 (coordination sequence for square lattice).
%C A008574 Susceptibility series H_1 for 2-dimensional Ising model (divided by 2).
%C A008574 Also the Engel expansion of exp^(1/4); cf. A006784 for the Engel expansion 
               definition - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 03 2002
%C A008574 Number of squares in an n X n board with all non-perimeter squares removed. 
               - Jon Perry (perry(AT)globalnet.co.uk), Jul 27 2003
%C A008574 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
%C A008574 Central terms of the triangle in A118013. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 10 2006
%C A008574 Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_4].
%D A008574 D. Hansel et al., Analytical properties of the anisotropic cubic Ising 
               model, J. Stat. Phys., 48 (1987), 69-80.
%D A008574 M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic 
               lattices, arXiv math.CO/0508136.
%H A008574 T. D. Noe, <a href="b008574.txt">Table of n, a(n) for n=0..1000</a>
%H A008574 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A008574 A. J. Guttmann, <a href="http://www.ms.unimelb.edu.au/~tonyg/articles/
               viennafinal.pdf">Indicators of solvability for lattice models</a>
               , Discrete Math., 217 (2000), 167-189.
%H A008574 S. Kitaev, <a href="http://www.integers-ejcnt.org/vol4.html">On multi-avoidance 
               of right angled numbered polyomino patterns</a>, Integers: Electronic 
               Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
%H A008574 S. Kitaev, <a href="http://www.ms.uky.edu/%7Emath/MAreport/4-ser.ps">
               On multi-avoidance of right angled numbered polyomino patterns</a>
               , University of Kentucky Research Reports (2004).
%F A008574 Binomial transform is A000337. - Paul Barry (pbarry(AT)wit.ie), Jul 21 
               2003
%F A008574 Euler transform of length 2 sequence [ 4, -2]. - Michael Somos Apr 16 
               2007
%F A008574 G.f.: ((1+x)/ (1-x))^2. E.g.f.: 1 +4*x*exp(x). - Michael Somos Apr 16 
               2007
%F A008574 a(-n)= -a(n) unless n=0. - Michael Somos Apr 16 2007
%F A008574 Row sums of triangle A130323: (1; 3,1; 5,2,1; 7,3,1,1;...). - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), May 24 2007
%F A008574 Row sums of triangle A131032: (1; 3,1; 5,2,1; 7,2,2,1;...). - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Jun 10 2007
%F A008574 G.f.: exp(4*atanh(x)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Oct 20 2009]
%o A008574 (PARI) {a(n)= 4*n+!n} /* Michael Somos Apr 16 2007 */
%Y A008574 Cf. A054275, A054410, A054389, A054764.
%Y A008574 Convolution square of A040000.
%Y A008574 Cf. A130323.
%Y A008574 Cf. A131032.
%Y A008574 Sequence in context: A161352 A008586 A059558 this_sequence A085127 A059532 
               A034045
%Y A008574 Adjacent sequences: A008571 A008572 A008573 this_sequence A008575 A008576 
               A008577
%K A008574 nonn,nice,easy
%O A008574 0,2
%A A008574 N. J. A. Sloane (njas(AT)research.att.com).