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.
Readers with more original suggestions for naming these numbers can always send their proposals to me at my e-mail address.

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 I compiled more than 70 Palindromic Quasi_Over_Squares.

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

Actually the following conjecture was proven for me by Neo Chee Beng and Warut Roonguthai almost simultaneously :

        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 ]

And also that :

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.

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online : Neil Sloane's Integer Sequences One can find the regular numbers of the form n2+(n+1) at %N Central polygonal numbers: n^2 – n + 1 under A002061. The palindromic numbers of the form n2+(n+1) are categorised as follows : %N n^2 + (n+1) is a palindrome under A028413. %N Palindromes of the form n^2 + (n+1) under A028414. 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.

If one searches the database for the short sequence {1,3,7,13} you get a positive reply and one of the entries shown is :

n(n – 1) + 1 =

(m+1)((m+1) – 1) + 1 =

(m+1)(m) + 1 =

m^2 + m + 1 =
m^2 + (m+1) QED

And the following phenomenon totally surprised me !
All the 'least significant' terms of the above pattern when multiplied with their reversals produce palindromes :

My program searched for palindromes exhaustively upto  length 31.

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(All rights reserved) - Last modified : May 29, 2007.

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

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[ TOP OF PAGE]

(All rights reserved) - Last modified : May 29, 2007.

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

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