login
Number of n-step walks on a cube lattice starting from the origin but not returning to it at any stage.
0

%I #8 Apr 19 2019 09:30:21

%S 1,6,30,180,1026,6156,35940,215640,1271106,7626636,45182124,271092744,

%T 1610875836,9665255016,57546367704,345278206224,2058613385346,

%U 12351680312076,73717606430364,442305638582184,2641804748619732

%N Number of n-step walks on a cube lattice starting from the origin but not returning to it at any stage.

%C a(n)/6^n tends to 0.65946...

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.

%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/polya/polya.html">Polya's Random Walk Constants</a> [Broken link]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010603070928/http://www.mathsoft.com/asolve/constant/polya/polya.html">Polya's Random Walk Constants</a> [From the Wayback machine]

%F a(2n) = 6*a(2n-1)-A049037(n); a(2n+1) = 6*a(2n).

%e a(2) = 30 since there are 36 2-step walks but 6 of them involve a return to the origin at some stage; similarly a(3) = 180 since there are 216 3-step walks but 36 of them involve a return to the origin at some stage.

%K nonn

%O 0,2

%A _Henry Bottomley_, Aug 28 2001