The initially palindromic numbers 1, 121, 12321, 1234321, 123454321, ... (OEIS A002477). For the first through ninth terms, the sequence is given by the generating function
 |
(1)
|
(Plouffe 1992, Sloane and Plouffe 1995).
The definition of this sequence is slightly ambiguous from the tenth term on, but the most common convention follows from the following observation. The sequences
of consecutive and reverse digits 
and 
, respectively, are given by
 |  |  |
(2)
|
 |  |  |
(3)
|
for 
, so the first few Demlo numbers
are given by
 |  |  |
(4)
|
 |  |  |
(5)
|
But, amazingly, this is just the square of the 
th repunit 
, i.e.,
 |
(6)
|
for 
, and the squares of the first
few repunits are precisely the Demlo numbers: 
, 
, 
, ... (OEIS A002275
and A002477). It is therefore natural to use
(6) as the definition for Demlo numbers 
with 
, giving 1, 121, ..., 12345678987654321, 1234567900987654321,
123456790120987654321, ....
The equality 
for 
also follows immediately from
schoolbook multiplication, as illustrated above. This follows from the algebraic
identity
 |
(7)
|
The sums of digits of the Demlo numbers for 
are given by
 |
(8)
|
More generally, for 
,
2, ..., the sums of digits are 1, 4, 9, 16, 25, 36, 49, 64, 81, 82, 85, 90, 97, 106,
... (OEIS A080151). The values of 
for which these are square are 1, 2, 3, 4, 5, 6, 7, 8, 9,
36, 51, 66, 81, ... (OEIS A080161), corresponding
to the Demlo numbers 1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321,
123456787654321, 12345678987654321, 12345679012345679012345679012345678987654320987654320987654320987654321,
... (OEIS A080162).
