Question 297613
For a {{{H=2}}} and {{{W=5}}} rectangle, you get 
10 {{{H=1}}},{{{W=1}}} rectangles 
8 {{{H=1}}}, {{{W=2}}} rectangles 
6 {{{H=1}}}, {{{W=3}}} rectangles 
4 {{{H=1}}}, {{{W=4}}} rectangles 
2 {{{H=1}}}, {{{W=5}}} rectangles 
5 {{{H=2}}}, {{{W=1}}} rectangles 
4 {{{H=2}}}, {{{W=2}}} rectangles 
3 {{{H=2}}}, {{{W=3}}} rectangles 
2 {{{H=2}}}, {{{W=4}}} rectangles 
1 {{{H=2}}}, {{{W=5}}} rectangles 
That's a total of {{{45}}} rectangles.
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Now for a formula.
Start simple and work up to see a pattern.
{{{H=2}}} , {{{W=1}}}
{{{1x1=2}}}
{{{2x1=1}}}
{{{3}}} total rectangles
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{{{ H=2}}}, {{{ W=2}}}
{{{ 1x1= 4}}}
{{{ 1x2= 2}}}
{{{ 2x1= 2}}}
{{{ 2x2= 1}}}
{{{ 9 }}}total rectangles.
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{{{ H=2}}}, {{{ W=3}}}
{{{ 1x1=6}}}
{{{ 1x2=4}}}
{{{ 1x3=2}}}
{{{ 2x1=3}}}
{{{ 2x2=2}}}
{{{ 2x3=1}}}
{{{ 18 }}}total
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{{{ H=2}}}, {{{ W=4}}}
{{{ 1x1=8}}}
{{{ 1x2=6}}}
{{{ 1x3=4}}}
{{{ 1x4=2}}}
{{{ 2x1=4}}}
{{{ 2x2=3}}}
{{{ 2x3=2}}}
{{{ 2x4=1}}}
{{{ 30}}} total rectangles.
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You're starting to get a pattern.
The {{{2xn}}} rectangles sum from {{{1}}} to {{{n}}} in steps of {{{1}}} for a total of {{{n(n+1)/2}}}.
The {{{1xn}}} rectangles sum for {{{2}}} to {{{2n}}} in steps of {{{2}}} or {{{n(n+1)}}}. 
So the grand total of rectangles in a {{{2xn}}} rectangle would be 
{{{S(n)=n(n+1)/2+n(n+1)}}}
{{{S(n)=(3n/2)(n+1)}}}