A190587 - OEIS

%I #30 Jul 08 2022 11:49:31

%S 1,2,4,10,24,56,130,304,714,1678,3944,9276,21832,51408,121088,285288,

%T 672304,1584638,3735596,8807312,20766914,48970942,115487946,272371376,

%U 642404770,1515218012,3574025956,8430514614,19886678810,46911630678,110664280068,261060908326

%N Number of two-sided n-step prudent walks ending on the northeast corner of their box, avoiding two or more consecutive west steps and south steps.

%H Alois P. Heinz, <a href="/A190587/b190587.txt">Table of n, a(n) for n = 0..800</a>

%H Shanzhen Gao, 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.

%F D-finite with recurrence (n+1)*a(n) -5*n*a(n-1) +(9*n-11)*a(n-2) +(-11*n+23)*a(n-3) +3*(4*n-9)*a(n-4) +2*(-2*n+1)*a(n-5) +(3*n-2)*a(n-6) +(-3*n+28)*a(n-7) +2*(-n+4)*a(n-8) +2*(-n+8)*a(n-9) +2*(n-10)*a(n-10)=0. - _R. J. Mathar_, May 30 2014

%F Recurrence (of order 9): (n-4)*(n+1)*a(n) = n*(4*n - 15)*a(n-1) - (5*n^2 - 24*n + 25)*a(n-2) + (6*n^2 - 33*n + 38)*a(n-3) - (6*n^2 - 30*n + 31)*a(n-4) - (2*n^2 - 24*n + 59)*a(n-5) - (5*n^2 - 42*n + 93)*a(n-6) - 2*(n^2 - 6*n + 14)*a(n-7) - 10*a(n-8) + 2*(n-9)*(n-3)*a(n-9). - _Vaclav Kotesovec_, Sep 03 2014

%F G.f.: ((1-x)*sqrt((1-x-x^3)^2 - 4*x^4) - 1 + 3*x-x^2+x^3 + x^4)/(x*(1-2*x-2*x^3)), see sequence 12 in link. - _Michel Marcus_, May 06 2015

%p b:= proc(d, i, n, x, y) option remember;

%p       `if`(n=0, `if`(x=0 and y=0, 1, 0),

%p          `if`(d<>3, b(1, x=0, n-1, max(x-1, 0), y), 0) +

%p          `if`(d<>4, b(2, y=0, n-1, x, max(y-1, 0)), 0) +

%p          `if`(d=0 or d=2 and i, b(3, false, n-1, x+1, y), 0) +

%p          `if`(d=0 or d=1 and i, b(4, false, n-1, x, y+1), 0))

%p     end:

%p a:= n-> b(0, false, n, 0, 0):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 09 2011

%t CoefficientList[Series[((1 - x) Sqrt[(1 - x - x^3)^2 - 4 x^4] - 1 + 3 x - x^2 + x^3 + x^4)/(x (1 - 2 x - 2 x^3)), {x, 0, 50}], x] (* _Vincenzo Librandi_, May 07 2015 *)

%K nonn,walk

%O 0,2

%A _Shanzhen Gao_, May 13 2011

%E More terms from _Alois P. Heinz_, Jun 09 2011