# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a006778 Showing 1-1 of 1 %I A006778 M2652 #25 Apr 05 2022 08:22:55 %S A006778 1,3,7,15,31,59,110,198,347,592,997,1641,2666,4266,6741,10525,16268, %T A006778 24882,37717,56683,84504,125031,183716,268125,388873,560647,803723, %U A006778 1146013,1625731,2294964,3224588,4510552,6282295,8714035,12039319,16570278,22723025 %N A006778 Number of n-step spirals on hexagonal lattice. %C A006778 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. %C A006778 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 %D A006778 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A006778 Sean A. Irvine, Java program (github). %H A006778 G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2 %H A006778 G. Szekeres and A. J. Guttmann, Spiral self-avoiding walks on the triangular lattice, J. Phys. A 20 (1987), 481-493. %Y A006778 Cf. A006776, A006777. %K A006778 nonn %O A006778 1,2 %A A006778 _N. J. A. Sloane_ %E A006778 More terms from _Sean A. Irvine_, Apr 04 2022 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE