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%I #29 Oct 24 2016 08:46:54
%S 1,3,6,15,35,83,195,460,1085,2560,6039,14247,33613,79306,187114,
%T 441477,1041626,2457630,5798569,13681202,32279488,76160166,179691649,
%U 423961718,1000285928,2360046161,5568211498,13137414580,30995819288,73129978187,172538870438
%N Number of two-sided n-step prudent walks ending on the top side of their box, avoiding two or more consecutive west steps and south steps.
%H Alois P. Heinz, <a href="/A190586/b190586.txt">Table of n, a(n) for n = 0..1000</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 Recurrence: (n+1)*(7*n^2 - 194*n + 926)*a(n) = n*(42*n^2 - 1143*n + 5320)*a(n-1) - (84*n^3 - 2349*n^2 + 13043*n - 13734)*a(n-2) + (77*n^3 - 2267*n^2 + 15130*n - 27824)*a(n-3) - (63*n^3 - 1893*n^2 + 13180*n - 24672)*a(n-4) - (7*n^3 - 313*n^2 + 3298*n - 8548)*a(n-5) + (77*n^3 - 2309*n^2 + 16800*n - 34964)*a(n-6) + 2*(14*n^3 - 563*n^2 + 6962*n - 24347)*a(n-7) + (49*n^3 - 1456*n^2 + 10086*n - 21316)*a(n-8) - (7*n^3 - 208*n^2 + 1687*n - 2950)*a(n-9) - 6*(7*n^3 - 236*n^2 + 2187*n - 5968)*a(n-10) - 2*(7*n^3 - 250*n^2 + 2526*n - 7744)*a(n-11) + 2*(n-10)*(7*n^2 - 180*n + 739)*a(n-12). - _Vaclav Kotesovec_, Sep 03 2014
%F G.f.: 1/(2*t*(1-2*t-t^2+t^3)*(1-2*t-2*t^3))*((1-2*t)*(1-t)*sqrt((1-t-t^3)^2-4*t^4)-(1+t)*(1-7*t+14*t^2-11*t^3+10*t^4-4*t^5)), see sequence 11 in link. - _Michel Marcus_, May 06 2015
%e a(2) = 6 since there are 6 such walks: WN, NW, EN, NE, EE, NN.
%p b:= proc(d, i, n, x, y) option remember;
%p `if`(n=0, `if`(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/(2 x (1-2 x - x^2 + x^3) (1 - 2 x - 2 x^3)) ((1 - 2 x) (1 - x) Sqrt[(1 - x - x^3)^2 - 4 x^4] - (1 + x) (1 - 7 x + 14 x^2 - 11 x^3 + 10 x^4 - 4 x^5)), {x, 0, 50}], x] (* _Vincenzo Librandi_, May 07 2015 *)
%t b[d_, i_, n_, x_, y_] := b[d, i, n, x, y] = If[n == 0, If[y == 0, 1, 0], If[d != 3, b[1, x == 0, n-1, Max[x-1, 0], y], 0] + If[d != 4, b[2, y == 0, n-1, x, Max[y-1, 0]], 0] + If[d == 0 || d == 2 && i, b[3, False, n-1, x+1, y], 0] + If[d == 0 || d == 1 && i, b[4, False, n-1, x, y+1], 0]]; a[n_] := b[0, False, n, 0, 0]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 24 2016, after _Alois P. Heinz_ *)
%K nonn,walk
%O 0,2
%A _Shanzhen Gao_, May 13 2011
%E More terms from _Alois P. Heinz_, Jun 09 2011
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