A006778 Number of n-step spirals on hexagonal lattice. (Formerly M2652)

1, 3, 7, 15, 31, 59, 110, 198, 347, 592, 997, 1641, 2666, 4266, 6741, 10525, 16268, 24882, 37717, 56683, 84504, 125031, 183716, 268125, 388873, 560647, 803723, 1146013, 1625731, 2294964, 3224588, 4510552, 6282295, 8714035, 12039319, 16570278, 22723025

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 III single spiral of Table 3 in Szekeres and Guttmann. In Model III every step of the walk consists of continuing in the current direction, turning clockwise by 120 degrees, or turning clockwise by 60 degrees. 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. If the angle turned is 120 degrees, then an extra u(j)=0 is inserted into the u sequence at that point. Then a walk with k straight line segments (including 0's as described), 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..37.

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. A006776, A006777.

Sequence in context: A229006 A023424 A276647 * A007574 A034480 A218281

Adjacent sequences: A006775 A006776 A006777 * A006779 A006780 A006781

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Apr 04 2022

Status

approved