|
%I #29 Jun 28 2021 08:48:54
%S 1,3,7,17,40,96,229,547,1306,3119,7448,17786,42473,101426,242206,
%T 578390,1381200,3298317,7876408,18808927,44915872,107259471,256136497,
%U 611656057,1460639684,3488019553,8329419319,19890721694,47499206650
%N Number of one-sided n-step prudent walks, avoiding 4 or more consecutive east steps.
%C a(n,k) is the number of one-sided n-step prudent walks, avoiding k or more consecutive east steps; k=4 in this sequence.
%H Alois P. Heinz, <a href="/A190360/b190360.txt">Table of n, a(n) for n = 0..1000</a>
%H Shanzhen Gao and Keh-Hsun Chen, <a href="http://worldcomp-proceedings.com/proc/p2014/FCS2696.pdf">Tackling Sequences From Prudent Self-Avoiding Walks</a>, FCS'14, The 2014 International Conference on Foundations of Computer Science.
%H S. Gao and H. Niederhausen, <a href="http://math.fau.edu/Niederhausen/HTML/Papers/Sequences%20Arising%20From%20Prudent%20Self-Avoiding%20Walks-February%2001-2010.pdf">Sequences Arising From Prudent Self-Avoiding Walks</a>, 2010.
%F G.f.: (1+t-t^k)/(1-2*t-t^2+t^(k+1)), (k=4 in this sequence).
%p b:= proc(n, i) option remember; `if`(n<0, 0,
%p `if`(n=0, 1, b(n-1,0) +`if`(i<=0, b(n-1,-1), 0)+
%p `if`(i>=0 and i<3, b(n-1,i+1), 0)))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 04 2011
%t (1+t-t^k)/(1-2*t-t^2+t^(k+1)) /. k -> 4 + O[t]^25 // CoefficientList[#, t]& (* _Jean-François Alcover_, Oct 24 2016 *)
%Y Cf. A006356 = a(n,2), A033303 = a(n,3).
%K nonn,walk
%O 0,2
%A _Shanzhen Gao_, May 09 2011
|
