Palindromic Sums of Powers of Consecutive Integers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only :

basen + ( base + 1 )n+1 + ...

Index 33 shows a particularly beautiful number in its most significant part.

Index 45 shows a palindromic basenumber in its most significant part.

Sources Revealed

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The normal numbers of the form n + (n+1)^2 are already categorised. %N Numbers of the form n + (n+1)^2 under A028387 . Check out the following two entries about Palindromic Powers %N n + (n+1)^2 is a palindrome under A028348. %N Palindromes of form n + (n+1)^2 under A028349. 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 {5,11,19,29,41} you get a positive reply and the entry shown is :

I will give a proof that every number of the form n^2 – n – 1 is also a member of the numbers of the form n + (n+1)^2
whereby its basenumbers are two units displaced.
Substitute n with m+2 and we derive :

n^2 – n – 1 [or n^2 – (n+1)] =

(m+2)^2 – (m+2) – 1 =

(m^2 + 4m + 4) – m – 3 =

m^2 + 3m + 1 =

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

n^2 + (n+1) =

(m+1)^2 + ((m+1)+1) – 2 =

(m^2+2m+1)+(m+2) – 2 =

m^2+3m+1 =

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

Not long after I submitted the numbers of the form n + (n+1)^2 under A028387, Neil Sloane pointed out to me that his Integer Sequence A028392 is the complement of the former. This means that what's missing in the first sequence appears in the second. The formula of the complement series is n + [sqrt(n)]. The square brackets indicate that the integer part of the real number in it must be considered.

When one subtracts the second part [sqrt(n)] from n ( See John Mellor's entry A028391 ) then another curiosity shows up that also relates n – [sqrt(n)] to the numbers of the form n + (n+1)^2. Each number preceding the couple of two identical numbers is also of the form n + (n+1)^2. See for yourself :

One infinite expanding patterns immediately shows up :

Palindromes Quasi_Under_Squares can only end with one of the following digits 1, 5 or 9 !

There exist no such palindromes of length 8, 16, 20.

My search program searched for palindromes exhaustively upto  length 33

The Table

In case you are also interested in the primes of these numberforms
check out the following entries in Sloane's table :

There are no primes of the form n + (n+1)^2 + (n+2)^3.
All such numbers are divisible by (n+3) twice !
The polynomial can be written as (n+1)*(n+3)*(n+3).

There are no primes of the form n + (n+1)^2 + (n+2)^3 + (n+3)^4.
All such numbers are even !

Contributions

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(All rights reserved) - Last modified : April 4, 2005.

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

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