Numbers with "perfect" figurate representation

[David Wilson]

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

----- Original Message ----- 
From: "Andrew Plewe" <aplewe at sbcglobal.net>
To: "Sequence Fans" <seqfan at ext.jussieu.fr>
Sent: Friday, March 28, 2008 3:21 PM
Subject: Numbers with "perfect" figurate representation


>I propose a sequence composed of numbers which have a
> "perfect" figurate representation. I think an
> illustration would explain it better than words, so
> here is an example:
>
> a(1) = 77:
>
> 11 14 17 20
> 11 13 15 17
> 11 12 13 14
> 11 11 11 11
> 11 10  9  8
> 11  9  7  5
> 11  8  5  2
>
> The representation is "perfect" because the sequences
> in the columns (i.e. (8,9,10,11,12,13,14),
> (5,7,9,11,13,15,17), etc.) match those centered around
> the middle row. So I can start at "5" in the last
> column and read leftwards (5,7,9,11) and continue with
> the row starting (11,13,15,17). No numbers in the
> representation can be either zero or negative. For
> contrast, here is an imperfect representation:
>
> n = 55:
>
> 11 13 15 17 19
> 11 12 13 14 15
> 11 11 11 11 11
> 11 10  9  8  7
> 11  9  7  5  3
>
> Here the rows don't match the columns so this
> representation isn't "perfect". Working by hand I've
> found the following numbers have a "perfect"
> representation:
>
> 25, 77, 153, 297, 533
>
> I have a suspicion that this sequence is a.)
> incomplete (i.e., there are more terms < 533 and >
> 533), and b.) probably covered by some other sequence
> in the OEIS. Can anyone confirm if either of those are
> true?
>
>    -Andrew Plewe-
>
>
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