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Palindromic Quasi_Over_Squares of the form n^2+(n+1)
n(n+0) n(n+1) n(n+2) n(n+x) n^2+1 n^2+x n^2–x n^2+(n+x)

  left to right (forwards) as from the right to left (backwards)

Quasi_Over_Square numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base2 + ( base + 1 )1

Let me name the numbers of the form n + (n+1)^2 or (n-1) + n^2 as follows : Quasi_Under_Squares.
Numbers of the form n^2 + (n+1) then become the Quasi_Over_Squares (coded as QO-SQU) .
Readers with more original suggestions for naming these numbers can always send their proposals to me at my e-mail address.

Just like Warut Roonguthai ( See Palindromic Pronic Numbers n(n+1) ) I also recognized
a similar finite growing pattern in these Quasi_Over_Squares basenumbers.

26 (5445)_n 62, for n = 2 to 4
26545 (5445)_n 437, for n = 0 to 4
|Astonishing!| the largest known Quasi_Over_Square now has a length of 47 digits !

 So far this compilation counts 160 Palindromic Quasi_Over_Squares.

All palindromic numbers of the form n^2 + (n+1) can only end with 1, 3 or 7

There are no palindromic quasi_over_squares of type x^2 + (x + 1) of EVEN length. All ODD lengths ⩽ 55 are represented.

A B C →  1 1 1 

It is no coincidence that I didn't find Palindromic Quasi_Over_Squares of even length.

        n mod 11 :   0   1   2   3   4   5   6   7   8   9   10
(n^2+n+1) mod 11 :   1   3   7   2  10   9  10   2   7   3    1
[ if zero appeared in the second line then it would be divisible by 11 ]

Either n or n+1 is even, so their product n(n+1) is even.
Therefore n(n+1)+1 is odd !!
Simple isn't it.

n(n – 1) + 1 =

(m+1) ( (m+1) – 1) + 1 =

(m+1) (m) + 1 =

m^2 + m + 1 =

m^2 + (m+1) QED

A lot of the palindromes are semiprimes. Is there a logical explanation for this phenomenon ?

 [9]
F_qs  30.103 =
      PRIME

Comm  30103 is the only known multi-digit palindromic prime 
      of the form sigma(n)/d(n), when n is composite. ( McCranie and Honaker )
      sigma(n) = the sum of all the factors of n, including 1 and n itself.
      d(n) = the number of factors of n, including 1 and n itself.

      30103 is also nice because it comes up for
      n = 149645 (an odd number) and once for
      n = 179574 (an even number).

      n         sigma(n)  d(n)  sigma(n)/d(n)=palprime
      149645    180618    6     30103
      179574    361236    12    30103

      30103 is still the only "known" palprime with the property.

PDG's program searched for palindromes exhaustively up to length 31.

From length 33 on these numbers were discovered by Feng Yuan during [ January 19-25, 2008 ].

[ November 4, 2022 ]
Missing entry [97] was spotted and added by Robert Xiao.

[ November 11, 2022 ]
Missing entries [105], [115] and [116] were added by Robert Xiao.

[ November 13, 2022 ]
Missing entries [124] was added by Robert Xiao.

[ November 14, 2022 ]
Seven new entries [135] up to [141] were discovered by David Griffeath using Rust code by Robert Xiao.

[ June 20, 2023 ]
Nine new entries [142] up to [150] were added by Patrick De Geest using CUDApalin code written by Robert Xiao.

[ July 26, 2023 ]
Five new entries [151] up to [155] were added by Patrick De Geest using CUDApalin code written by Robert Xiao.

[ October 6, 2024 ]
Five new entries [156] up to [160] were added by Patrick De Geest using CUDApalin code written by Robert Xiao.

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E-mail address : pdg@worldofnumbers.com