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

1, 4, 4, 12, 8, 20, 12, 28, 16, 36, 20, 44, 24, 52, 28, 60, 32, 68, 36, 76, 40, 84, 44, 92, 48, 100, 52, 108, 56, 116, 60, 124, 64, 132, 68, 140, 72, 148, 76, 156, 80, 164, 84, 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

Offset: 0

Author

N. J. A. Sloane, Mar 23 2025

Keywords

• nonn

Comments

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).

References

• 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).

Links

• 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).

Crossrefs

Partial sums give A319384.

Cf. A382155, A382156.

Programs

Mathematica

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

Formula

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