Truncated Pyramid

I was surprised to discover that the number associated with my diurnal age today (27930) has a connection to the volume of a truncated square pyramids. Firstly, let's recall the formula for the volume of such figure:$$V=\frac{1}{3} (a^2+ab+b^2) \, h$$where \(a\) is the side length of the square base, \(b\) is the side length of the top square and \(h\) is the height of the truncated pyramid. The formula is easily confirmed by integration from first principles without relying on the formula for the volume of a pyramid. 

Now 27930 is a member of OEIS A027444: \( \text{a}(n) = n^3 + n^2 + n\) with the following interesting comment attached to the entry:

For \(n>1\), a(\(n\)) is the volume of a truncated square pyramid with height \(n\) and base  lengths \(n+2\) and \(n-1\). See Figure 1.

Figure 1

Now if we substitute \(a=n+2\), \(b=n-1\) and \(n=h\) into our earlier formula we find the volume \(V\) of the truncated pyramid becomes:$$V=n^3+n^2+n$$When \(n=30\), the volume is 27930 cubic units. Here are the volumes for various values of \(n>1\):

  n    n-1   n+2   V 

  2    1     4     14

  3    2     5     39

  4    3     6     84

  5    4     7     155

  6    5     8     258

  7    6     9     399

  8    7     10    584

  9    8     11    819

  10   9     12    1110

  11   10    13    1463

  12   11    14    1884

  13   12    15    2379

  14   13    16    2954

  15   14    17    3615

  16   15    18    4368

  17   16    19    5219

  18   17    20    6174

  19   18    21    7239

  20   19    22    8420

  21   20    23    9723

  22   21    24    11154

  23   22    25    12719

  24   23    26    14424

  25   24    27    16275

  26   25    28    18278

  27   26    29    20439

  28   27    30    22764

  29   28    31    25259

  30   29    32    27930

  31   30    33    30783

  32   31    34    33824

  33   32    35    37059