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Palindromic Incremented Squares of Form n^2+1
n(n+0) n(n+1) n(n+2) n(n+x) n^2+x n^2–x n^2+(n+1) n^2+(n+x)

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

Palindromic Incremented Squares are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base2 + 1

So far this compilation counts 117 Palindromic Incremented Squares (coded as SQUADD1).

Palindromic Incremented Squares can only start or end with digits : 0, 1, 2, 5, 6 or 7.

There are no palindromic incremented squares of type [SQ1] of EVEN lengths.

A B C →  1 0 1 

10ⁿ n = 0, 1 , 2, ... 1, 10, 100, 1000, ...

10³ⁿ+2*10ⁿ n = 1, 2, 3, ... 1020, 1000200, 1000002000, ...

10³ⁿ⁺¹+9*10ⁿ n = 1, 2, 3, ... 10090, 10000900, 10000009000, ...

10⁵ⁿ+2*10³ⁿ+2*10ⁿ n = 1, 2, 3, ... 102020, 10002000200, 1000002000002000, ...

100000 10ⁿ⁺¹ n = 1, 2, 3, ...

10000000090000 10³ⁿ⁺¹+9*10ⁿ n = 1, 2, 3, ...

1000000009000000090000 10⁵ⁿ⁺¹+9*10³ⁿ+9*10ⁿ n = 1, 2, 3, ...

100000000900000009000000090000 10⁷ⁿ⁺¹+9*10⁵ⁿ+9*10³ⁿ+9*10ⁿ n = 1, 2, 3, ...

10000000090000000900000009000000090000 10⁹ⁿ⁺¹+9*10⁷ⁿ+9*10⁵ⁿ+9*10³ⁿ+9*10ⁿ n = 1, 2, 3, ...

1000000009000000090000000900000009000000090000 10¹¹ⁿ⁺¹+9*10⁹ⁿ+9*10⁷ⁿ+9*10⁵ⁿ+9*10³ⁿ+9*10ⁿ n = 1, 2, 3, ...

Warut Roonguthai from Bangkok Thailand informed me that every (palindromic) number of the form n(n+2)
is also of the form n^2–1.

" It's just one step away from being a palindromic square. And
that is why I think that it is interesting to investigate palindromes
of the form n^2+1, another near miss, as well. "

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

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The regular incremented squares are categorised as follows : %N n^2 + 1 under A002522. Soon the following two entries about Palindromic Incremented Squares will be present : %N n^2 + 1 is a palindrome under A027719. %N Palindromes of form n^2 + 1 under A027720 Thanks N. Sloane for providing the palindromic entry number 027720 Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

My search went exhaustively up to (only odd!) length 33.

[ November 3, 1997 ]
Warut Roonguthai (email) from Bangkok (Thailand) helped searching for these Palindromic Incremented Squares.
He discovered those starting from index number [21] up to [41] and [48] was the largest sporadic one which he submitted to me.

From length 35 on these numbers were discovered by Feng Yuan during [ January 20-25, 2008 ].

[ November 4, 2022 ]
Robert Xiao (email) spotted an error in entry [50] and has now been fixed.
Furthermore he added two missing entries [69] and [70].

[ November 11, 2022 ]
Robert Xiao (email) added missing entry [81].

[ November 13, 2022 ]
Robert Xiao (email) added four new entries [95] up to [98].

[ December 6, 2022 ]
Robert Xiao (email) added nine new entries [99] up to [107].

[ June 17, 2023 ]
Patrick De Geest (email) added nine new entries [108] up to [117].

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( © All rights reserved ) - Last modified : July 26, 2023.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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