OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..750
Mireille Bousquet-Mélou, Families of prudent self-avoiding walks, DMTCS proc. AJ, 2008, 167-180.
Mireille Bousquet-Mélou, Families of prudent self-avoiding walks, arXiv:0804.4843 [math.CO], 2008-2009.
Enrica Duchi, On some classes of prudent walks, in: FPSAC'05, Taormina, Italy, 2005.
FORMULA
G.f.: (2*(1-t^2)*(1-t)*U/((1-t*U)*(2*t-U)))-1 with U = (1-t+t^2+t^3 -sqrt((1-t^4)*(1-2*t-t^2)))/(2*t).
a(n) ~ (1 - 3*r + (r-1)*sqrt(-7+8*r+24*r^2)) * (-40-22*r+35*r^2) / (37*r^n), where r = 0.40303171676268... is the root of the equation 1 - 2*r - 2*r^2 + 2*r^3 = 0. - Vaclav Kotesovec, Sep 10 2014
Conjecture: +(n+1)*a(n) -4*n*a(n-1) +(n-5)*a(n-2) +2*(4*n-3)*a(n-3) +3*(-n+3)*a(n-4) +2*(n-8)*a(n-5) +(-n+13)*a(n-6) +8*(-n+7)*a(n-7) +2*(n-7)*a(n-8) +2*(n-9)*a(n-9)=0. - R. J. Mathar, Sep 16 2017
EXAMPLE
a(4) = 24: EEEE, EEEN, EENE, EENN, ENEE, ENEN, ENNE, ENNN, ESEN, NEEE, NEEN, NENE, NENN, NNEE, NNEN, NNNE, NNNN, NWNE, WNEE, WNEN, WNNE, SEEN, SENE, SENN.
MAPLE
U:= (1-t+t^2+t^3 -sqrt((1-t^4)*(1-2*t-t^2)))/(2*t):
gf:= (2*(1-t^2)*(1-t)*U/((1-t*U)*(2*t-U))) -1:
a:= n-> coeff(series(gf, t, n+4), t, n):
seq(a(n), n=0..30);
MATHEMATICA
U = (1 - t + t^2 + t^3 - Sqrt[(1 - t^4)(1 - 2t - t^2)])/(2t);
gf = (2(1 - t^2)(1 - t) U/((1 - t U)(2t - U))) - 1;
a[n_] := SeriesCoefficient[gf, {t, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 24 2018, from Maple *)
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Alois P. Heinz, Jun 09 2011
STATUS
approved