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A127617 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 3 with the steps (1,0), (0, 1), (2,0) and (0,2).

%I A127617 #17 Sep 08 2022 08:45:29
%S A127617 1,1,5,22,92,395,1684,7189,30685,130973,559038,2386160,10184931,
%T A127617 43472696,185556025,792015257,3380586357,14429474710,61589830404,
%U A127617 262886022219,1122085581740,4789437042413,20442921249973,87257234103245,372443097062686,1589711867161816
%N A127617 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 3 with the steps (1,0), (0, 1), (2,0) and (0,2).
%H A127617 Muniru A Asiru, <a href="/A127617/b127617.txt">Table of n, a(n) for n = 0..1550</a>
%H A127617 Arvind Ayyer and Doron Zeilberger, <a href="https://arxiv.org/abs/math/0610734">The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points</a>, arXiv:math/0610734 [math.CO], 2006-2007.
%H A127617 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,5,2,-1).
%F A127617 G.f.: (1 - 2*x - 3*x^2) / (1 - 3*x - 5*x^2 - 2*x^3 + x^4).
%F A127617 a(n) = 3*a(n-1)+5*a(n-2)+2*a(n-3)-a(n-4). - _Vincenzo Librandi_, Dec 13 2018
%e A127617 a(2)=5 because we can reach (2,2) in the following ways:
%e A127617 (0,0),(1,0),(1,1),(2,1),(2,2)
%e A127617 (0,0),(2,0),(2,2)
%e A127617 (0,0),(1,0),(2,0),(2,2)
%e A127617 (0,0),(2,0),(2,1),(2,2)
%e A127617 (0,0),(1,0),(2,0),(2,1),(2,2)
%p A127617 seq(coeff(series((1-2*x-3*x^2)/(1-3*x-5*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Dec 10 2018
%t A127617 LinearRecurrence[{3, 5, 2, -1}, {1, 1, 5, 22}, 23] (* _Jean-François Alcover_, Dec 10 2018 *)
%t A127617 CoefficientList[Series[(1 - 2 x - 3 x^2) / (1 - 3 x - 5 x^2 - 2 x^3 + x^4), {x, 0, 33}], x] (* _Vincenzo Librandi_, Dec 13 2018 *)
%t A127617 b[n_, k_] := Boole[n >= 0 && k >= 0 && 0 <= n-k <= 3];
%t A127617 T[0, 0] = T[1, 1] = 1; T[n_, k_] /; b[n, k] == 1 := T[n, k] = b[n-2, k]* T[n-2, k] + b[n-1, k]*T[n-1, k] + b[n, k-2]*T[n, k-2] + b[n, k-1]*T[n, k-1]; T[_, _] = 0;
%t A127617 a[n_] := T[n, n];
%t A127617 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 03 2019 *)
%o A127617 (Magma) I:=[1,1,5,22]; [n le 4 select I[n] else 3*Self(n-1)+5*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 13 2018
%Y A127617 Cf. A000108, A046717, A122951, A127618, A127619, A127620.
%K A127617 nonn,easy,walk
%O A127617 0,3
%A A127617 _Arvind Ayyer_, Jan 20 2007