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Palindromic Quasipronic Numbers of the form n(n+2)
n(n+0) n(n+1) n(n+x) n^2+1 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 Products of Two Sequential Integers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base x ( base + n )

In case of n = 2 note that the formula n(n+2) can be written also as n²+2n.
Its always easier to refer something by name so I'll baptise these numbers Palindromic Quasipronic Numbers.
A repetitive infinite palindromic pattern can be made in the following manner :

9 x 11 = 99
99 x 101 = 9999
999 x 1001 = 999999
9999 x 10001 = 99999999

So far I compiled 130 Palindromic Quasipronic Numbers (coded as Q-PRONIC).

Palindromic Quasipronic Numbers of the form x(x+2) can only start or end in one of the following digits : 0, 3, 4, 5, 8 or 9.

There are no such palindromic quasipronic numbers of lengths 35, 43.

A B C →  1 2 0 

Warut Roonguthai (email) 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. "

Proof that numbers of the form n^2–1 equal n(n+2) by substituting n with m+1 :

n^2 – 1 =

(m + 1)^2 – 1 =

m^2 + 2m + 1 – 1 =

m^2 + 2m =

m(m + 2)  QED

A finite palindromic pattern hides in the list in the following manner :

2 x 4 = 8
2966 x 2968 = 8803088
2967032 x 2967034 = 8803284823088
2967032966 x 2967032968 = 8803284627264823088
2967032967032 x 2967032967034 = 8803284627460647264823088

Note that the lengths of the resulting palindromes increases with steps of 6 !

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The regular quasipronic numbers are categorised as follows : %N n(n+2) under A005563. Check out the following two entries about Palindromic Quasipronic Numbers %N n(n+2) is a palindrome under A028503. %N Palindromes of form n(n+2) under A028504. %N Both n and its reverse are one less than a square under A066619. Click here to view some the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

PDG's program completed the search up to length 26 inclusive.

[ January 15, 2008 ]
From length 27 on these quasipronic numbers were submitted by Feng Yuan.

[ November 4, 2022 ]
Robert Xiao (email) added two missing entries [87] and [88].

[ November 11, 2022 ]
Robert Xiao (email) added one missing entry [102].

[ November 13, 2022 ]
Robert Xiao (email) added one missing entry [102]
and eight more new entries [109] up to [116].

[ December 6, 2022 ]
Robert Xiao (email) added six new entries [117] up to [122].

[ June 16, 2023 ]
Patrick De Geest (email) added eight new entries [123] up to [130].

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( © All rights reserved ) - Last modified : July 3, 2023.

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

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