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A382154 a(0) = 1; thereafter a(n) = 2*n if n even or 4*n if n odd.

%I A382154 #40 Mar 24 2025 13:51:53
%S A382154 1,4,4,12,8,20,12,28,16,36,20,44,24,52,28,60,32,68,36,76,40,84,44,92,
%T A382154 48,100,52,108,56,116,60,124,64,132,68,140,72,148,76,156,80,164,84,
%U A382154 172,88,180,92,188,96,196,100,204,104,212,108,220,112,228,116,236,120,244,124,252,128,260,132,268,136,276,140,284,144,292
%N A382154 a(0) = 1; thereafter a(n) = 2*n if n even or 4*n if n odd.
%C A382154 Let G denote the 2-dimensional grid obtained from the square grid Z X Z by deleting the vertices with both coordinates odd and the four edges at each of those vertices (see link). G has vertices with valency either 2 (one coordinate even and one odd, indicated by X) or 4 (both coordinates even, indicated by O). The present sequence is the coordination sequence of G with respect to a vertex of valency 4.
%C A382154 G arises in connection with the six-vertex lattice model of statistical mechanics (see Gorin-Nicoletti).
%D A382154 Gorin, Vadim, and Matthew Nicoletti. "Six-Vertex Model and Random Matrix Distributions," Bull. Amer. Math. Soc., 62:2 (2025), 175-234 (See Fig. 1.2).
%H A382154 Paolo Xausa, <a href="/A382154/b382154.txt">Table of n, a(n) for n = 0..10000</a>
%H A382154 Vadim Gorin and Matthew Nicoletti, <a href="https://arxiv.org/abs/2309.12495">Six-Vertex Model and Random Matrix Distributions</a>, arXiv:2309.12495 [math-ph], 2023-2024.
%H A382154 N. J. A. Sloane, <a href="/A382154/a382154.jpg">The grid G</a> (Each X-node is joined to two O-nodes, and each O-node to four X-nodes.)
%H A382154 N. J. A. Sloane, <a href="/A382154/a382154_1.jpg">Illustrates the initial terms of the coordination sequence of G with respect to a vertex of degree 4.</a> E.g. the 12 red vertices labeled 3 correspond to a(3) = 12.
%H A382154 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A382154 G.f.: (1+x^2)*(1+4*x+x^2)/(1-x^2)^2.
%t A382154 Join[{1}, Riffle[8*# - 4, 4*#]] & [Range[50]] (* _Paolo Xausa_, Mar 24 2025 *)
%o A382154 (Python)
%o A382154 def A382154(n): return n<<(1<<(n&1)) if n else 1 # _Chai Wah Wu_, Mar 24 2025
%Y A382154 Partial sums give A319384.
%Y A382154 Cf. A382155, A382156.
%K A382154 nonn
%O A382154 0,2
%A A382154 _N. J. A. Sloane_, Mar 23 2025