Eeven[n_]  =  (n - 1/2)^2;
Eodd[n_]  =  n^2;
evenlist  =  {Eeven[1], Eeven[2], Eeven[3], Eeven[4], Eeven[5], 
        Eeven[6]}
oddlist  =  {Eodd[1], Eodd[2], Eodd[3], Eodd[4], Eodd[5], 
        Eodd[6]}

allstates  =  Join[evenlist, oddlist]

Sort[allstates]

p1  =  
      ListPlot[evenlist, 
        PlotStyle -> {PointSize[0.02], 
            Hue[ .6]}]

p2  =  
      ListPlot[oddlist, 
        PlotStyle -> {PointSize[0.02], Hue[ .9]}]

Show[p1, p2]

even[x_, n_]  =  
      Cos[(n  -  1/2) Pi x]
odd[x_, n_]  =  Sin[n Pi x]

Plot[even[x, 1], {x, -1, 1}]
Plot[odd[x, 1] , {x, -1, 1}]

Plot[even[x, 2], {x, -1, 1}]
Plot[odd[x, 2] , {x, -1, 1}]

Plot[even[x, 3], {x, -1, 1}]
Plot[odd[x, 3] , {x, -1, 1}]