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A191625
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Number of n-step two-sided prudent self-avoiding walks ending at the northeast corner of their box.
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3
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1, 2, 4, 10, 24, 58, 140, 340, 828, 2022, 4948, 12130, 29780, 73202, 180124, 443614, 1093376, 2696622, 6654568, 16430016, 40583388, 100283298, 247890520, 612951190, 1516046060, 3750655206, 9281098840, 22970992024, 56864328080, 140790241078, 348635151944
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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);
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MATHEMATICA
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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}];
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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