SpiralFrogs.nb

(**First we make a function to obtain every spiral corner, then we interpolate between each corner to get the sides, sum together, and finally convert the bijection to an integer sequence.*))

up = {0, 1};
right = {1, 0};
down = -up;
left = -right;
directions = {up, right, down, left};
start = {0, 0};
length = 100;

corners = Table[
   start + Sum[Ceiling[i/2] directions[[Mod[i, 4, 1]]], {i, 1, n}],
   {n, 0, length}];

Interpolate[a_, b_] := 
  Table[a + i (b - a)/Norm[b - a], {i, 1, Norm[b - a]}];
spiral = Join[
   {{0, 0}},
   Join @@ 
    MapThread[Interpolate, {corners[[1 ;; -2]], corners[[2 ;;]]}]
   ];

ListLinePlot[spiral, PlotLabel -> "Spiral Bijection"]
FindFrog[f_, t_] := f[[1]] t + f[[2]];
frogSequence = MapThread[FindFrog, {spiral, Range[Length[spiral]]}];
ListPlot[frogSequence, PlotLabel -> "Integer Sequence"]

The output plots for convenience: