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A216212 Number of n step walks (each step +-1 starting from 0) which are never more than 4 or less than -4.

%I A216212 #30 Oct 19 2020 12:29:40
%S A216212 1,2,4,8,16,30,60,110,220,400,800,1450,2900,5250,10500,19000,38000,
%T A216212 68750,137500,248750,497500,900000,1800000,3256250,6512500,11781250,
%U A216212 23562500,42625000,85250000,154218750,308437500,557968750,1115937500,2018750000,4037500000
%N A216212 Number of n step walks (each step +-1 starting from 0) which are never more than 4 or less than -4.
%C A216212 The number of n step walks (each step +-1 starting from 0) which are never more than k or less than -k is given by a(n,k) = 2^n/(k+1)*Sum_{r=1..k+1} (-1)^r*cos((Pi*(2*r-1))/(2*(k+1)))^n*cot((Pi*(1-2*r))/(4*(k+1))), n<>0 if k even. Here we have k=4. - _Herbert Kociemba_, Sep 22 2020
%H A216212 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-5).
%F A216212 a(n) = A068913(4,n).
%F A216212 G.f.: (1+2*x-x^2-2*x^3+x^4)/(1-5*x^2+5*x^4).
%F A216212 a(n) = 5*a(n-2) - 5*a(n-4), a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, a(4) = 16.
%F A216212 a(2*n+1) = 2*A039717(n+1), a(2*n+2) = 4*A039717(n+1).
%F A216212 a(n) = (2^n/5)*Sum_{r=1..5} (-1)^r*cos(Pi*(2*r-1)/10)^n*cot(Pi*(1-2*r)/20), n>0. - _Herbert Kociemba_, Sep 22 2020
%t A216212 nn=30;CoefficientList[Series[(1+x-x^2)^2/(1-5x^2+5x^4),{x,0,nn}],x] (* _Geoffrey Critzer_, Jan 14 2014 *)
%t A216212 a[0,4]=1; a[n_,k_]:=2^n/(k+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (k+1))]^n Cot[(Pi (1-2r))/(4 (k+1))],{r,1,k+1}]
%t A216212 Table[a[n,4],{n,0,40}]//Round (* _Herbert Kociemba_, Sep 22 2020 *)
%Y A216212 Cf. A068911, A068912, A068913, A178381 (starting from 4).
%K A216212 nonn,walk
%O A216212 0,2
%A A216212 _Philippe Deléham_, Mar 13 2013