A006777 Number of n-step spirals on hexagonal lattice. (Formerly M1098)

1, 2, 4, 8, 14, 26, 43, 74, 120, 197, 311, 495, 768, 1189, 1811, 2748, 4116, 6136, 9058, 13299, 19370, 28069, 40399, 57856, 82374, 116736, 164574, 231007, 322749, 449089, 622263, 858935, 1181048, 1618209, 2209299, 3006273, 4077285, 5512650, 7430440, 9986147

Offset

1,2

Comments

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Corresponds to the Model II single spiral of Table 3 in Szekeres and Guttmann. In Model II every step of the walk consists of either continuing in the current direction or turning clockwise by 60 degrees (i.e., giving an obtuse angle on the path). Roughly speaking, a "single spiral" is a self-avoiding clockwise walk that cannot get stuck in a dead end. More precisely, let u(i) denote the length of the successive straight-line segment of the walk with u(0)=0. Then a walk with k straight line segments (note k <= n), is a single spiral if u(i-4) + u(i-3) < u(i-1) + u(i) for 4 <= i <= k. - Sean A. Irvine, Apr 05 2022

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..40.

Sean A. Irvine, Java program (github).

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

G. Szekeres and A. J. Guttmann, Spiral self-avoiding walks on the triangular lattice, J. Phys. A 20 (1987), 481-493.

Crossrefs

Cf. A309982, A006778.

Sequence in context: A298880 A208483 A284735 * A036609 A027557 A120545

Adjacent sequences: A006774 A006775 A006776 * A006778 A006779 A006780

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Apr 04 2022

Status

approved