Hello, Jazzy331!
A fascinating problem . . .
My problem is how to express a formula introducing which would explain the sequence:
. .
I think I found the H-formula for the Edges and Middles.
. .
. .
I'm having a problem understanding what must be an easy problem, but I can't see a way through it.
It relates to sequences in counting the number of corners, edges and middle panes in a square or rectangular of equal sized pieces. So with a 3x3 square, there are:
c e c............c - Corner panes
e m e........... e - Edge panes
c e c........... m - Middle panes
.......Corners...Edges...Middle
3x3.......4..........4.........1 - So it follows that:
3x4.......4..........6.........2 - If the width 'W' is variable, the sequence can be expressed as
3xW......4.......2w-2.....w-2
Doing the same for a 4 sides square:
.......Corners...Edges...Middle
4x4.......4..........8.........4 - So it follows that:
4x5.......4.........10........6 - If the width 'W' is variable, the sequence can be expressed as
4xW......4.........2w.....2w-4
My problem is how to express a formula when introducing a variable Height 'H' which would explain the sequence:
.......Corners...Edges...Middle
3xW.....4........2w-2.....w-2
4xW.....4.........2w......2w-4
5xW.....4........2w+2....3w-6
HxW....4..........??.........??
Any help would be appreciated. Thanks in advance.
Thanks Soroban - that's brilliant.
Is there a specific method of working to get me there?
I have followed the maths to test it and agree with the H-4 element on the Edges, but my problem with maths is always that I need to understand what I've done to get somewhere, otherwise it won't stick. Is it purely that the '1' in 5xW 2(w+1) resolves the h-4? If so, then I assume that the same would apply to the '3' in 5xW : 3(w-2) in the middles H-2 = 3.
But why does the expression of the formula work that way? Any thoughts?
Thanks in advance.
Jazzy331