This is a kind of formula math professionals appear to avoid... I had to try if I could turn it into a nice effect

3x+1.bb

Main
Const Cycles = 360
Const Width = 1200
Const Height = 1000




Function Even(a%)
	Return a Mod 2 = 0
End Function



Function Max(A,B)
	If A>B
		Return A
	Else
		Return B
	EndIf
End Function







Function Main()
	DebugLog "Let's do it!   "+Cycles+" Cycles"
	AppTitle "3x+1"
	Graphics Width,Height
	SetBuffer FrontBuffer()
	For ak=1 To Cycles
		DebugLog "Cycle: "+Ak
		Local R% = Abs(Sin(ak)*255)
		Local G% = Abs(Cos(ak)*255)
		Local B% = Abs(ak-Floor(Cycles/2)) Mod 256
		Color R,G,B
		Local D = ak
		For Pos=0 To Max(Width,Height)
			;DebugLog "Cycle: "+Ak+"; Pos: "+Pos
			If Even(D)
				D = D / 2
			Else
				D = (D*3) + 1
			EndIf
			If D>4
				DebugLog "Cycle: "+Ak+"; Pos: "+Pos+"; D: "+D
			EndIf
			For Y=0 To D
				; Base
				Plot Pos,Y
				Plot Pos,Height-Y
				Plot Y,Pos
				Plot Width-Y,Pos
				; Inverse
				Plot Width-Pos,Y
				Plot Width-Pos,Height-Y
				Plot Y,Height-Pos
				Plot Width-Y,Height-Pos				
			Next
		Next
	Next
	WaitKey 
	DebugLog "It's over"
	End
End Function

The result looks a bit like this