Ploting the expression $(nx + 1/2) + ny^2$

Question

I have the expression (nx + 1/2) + ny^2, where nx >= 0 and ny > 0, whee both nx and ny are integers.
I want to show and plot the lowest 10 values of the expression in way that shows the range of nx and ny.
How do I do it?

Bob Hanlon · Answer

Clear["Global`*"]

f[nx_, ny_] = (nx + 1/2) + ny^2;

data =
 SortBy[
   Flatten[
    Table[{nx, ny, f[nx, ny]}, {nx, 10}, {ny, 10}],
    1],
   Last][[1 ;; 10]]

(* {{1, 1, 5/2}, {2, 1, 7/2}, {3, 1, 9/2}, {1, 2, 11/2}, {4, 1, 11/
  2}, {2, 2, 13/2}, {5, 1, 13/2}, {3, 2, 15/2}, {6, 1, 15/2}, {4, 2, 
  17/2}} *)

nxRange = MinMax[data[[All, 1]]]

(* {1, 6} *)

nyRange = MinMax[data[[All, 2]]]

(* {1, 2} *)

ListPointPlot3D[
 Callout @@@ Transpose[{data, Range[10]}],
 Filling -> Axis,
 ColorFunction -> "Rainbow",
 AxesLabel -> (Style[#, 14, Bold] & /@ {"nx", "ny", "f"})]

ListPlot[Callout[Last[#], Most[#]] & /@ data,
 AxesLabel -> (Style[#, 14, Bold] & /@ {"Order", "f"}),
 ColorFunction -> "Rainbow"]

Ploting the expression $(nx + 1/2) + ny^2$

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