0,2

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.

G arises in connection with the six-vertex lattice model of statistical mechanics (see Gorin-Nicoletti).

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).

Paolo Xausa, Table of n, a(n) for n = 0..10000

Vadim Gorin and Matthew Nicoletti, Six-Vertex Model and Random Matrix Distributions, arXiv:2309.12495 [math-ph], 2023-2024.

N. J. A. Sloane, The grid G (Each X-node is joined to two O-nodes, and each O-node to four X-nodes.)

N. J. A. Sloane, Illustrates the initial terms of the coordination sequence of G with respect to a vertex of degree 4. E.g. the 12 red vertices labeled 3 correspond to a(3) = 12.

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

G.f.: (1+x^2)*(1+4*x+x^2)/(1-x^2)^2.

Join[{1}, Riffle[8*# - 4, 4*#]] & [Range[50]] (* Paolo Xausa, Mar 24 2025 *)

(Python)

def A382154(n): return n<<(1<<(n&1)) if n else 1 # Chai Wah Wu, Mar 24 2025

Partial sums give A319384.

Cf. A382155, A382156.

Sequence in context: A273742 A169710 A269629 * A241496 A394329 A273412

Adjacent sequences: A382151 A382152 A382153 * A382155 A382156 A382157

N. J. A. Sloane, Mar 23 2025

approved