Numbers with "perfect" figurate representation

[Andrew Plewe]

Perhaps I should clarify -- the representation must be "complete", i.e. 
you can't continue rightward without hitting zero or a negative number. 
  Hence some of the values you found violate the "column" rule -- i.e. 
that the sequence of numbers in the columns must match those in the 
rows. For example, n=15 doesn't work because the "complete" 
representation of n using 5 as the "base" is:

5 6 7 8 9
5 5 5 5 5
5 4 3 2 1

Around the perimeter the sequence is (1,2,3,4,5,6,7,8,9). However, the 
last row is (1,5,9). For the representation to be "perfect", the last 
column sequence should match that of the perimeter, such as for the 
number 25:

5 7 9
5 6 7
5 5 5
5 4 3
5 3 1

You get (1,3,5,7,9) whether you go around the perimeter or up the last 
column. In fact all the "row" sequences match their respective "column" 
sequences in such a representation. I'm re-evaluating the sequence you 
came up with; so far I've verified these values up to and including 30:

6,9,25,30

	-Andrew Plewe-


David Wilson wrote:
> In a perfect tableau, there must be as many rows above the central row 
> as below. Let r be this number of rows.
> 
> The leftward increment in the nth row below the center row is the same 
> as the upward increment in the nth column to the right of the first 
> column. For the increment in the last row to match the increment in the 
> last column, there must be as many rows below the center row as there 
> are columns to the right of the first column. This means that a perfect 
> tableau has width r+1 and height 2r+1.
> 
> Let a be the element repeated in the first column. The smallest element 
> in the tableau is then the lower right element, which is a-r^2. All the 
> elements in the tableau are positive iff this element is positive, which 
> happens when a > r^2.
> 
> The number associated with the tableau is (apparently) the column sum, 
> which is a(2r+1).
> 
> Therefore, your perfect tableau numbers are numbers of the form a(2r+1) 
> were a > r^2. The corresponding tableau has width r+1, height 2r+1, and 
> its first column and center row are filled with element a.
> 
> If we allow r = 0, then any positive integer a > r^2, so that a(2r+1) = 
> a is a perfect tableau number. This corresponds to the degenerate 1x1 
> perfect tableau with single element a. I assume you require r >= 1 to 
> avoid these degenerate tableaux.
> 
> Even if we do, perfect tableau numbers are quite common, including, for 
> instance, all 3n with n >= 2. The perfect tableau numbers < 100 are
> 
> 6 9 12 15 18 21 24 25 27 30 33 35 36 39 40 42 45 48 50 51 54 55 57 60
> 63 65 66 69 70 72 75 77 78 80 81 84 85 87 90 91 93 95 96 98 99 100