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[Alexandre Wajnberg]

Hi,

Eric Angelini asks me to send you his post, because since the arrival of the
new computing hardware in his company, his computer adress is not recognized
by some robots...

Alexandre

-------------------------

> 
> Hello SeqFan,
> 
> A « balanced word », according to Dmitri Borgmann (a longtime
> collaborator of Word Ways) can be divided in two parts of
> equal weight like this:
> 
>                   M . I . N . U . T . E
>                   5 4 3 2 1 0 1 2 3 4 5
>                   ----------^----------
> 
> The left side (MIN) has a weight of 106
> [sum of (rank of each letter x distance from 0)]:
> 
> Mx5 + Ix3 + Nx1 = 13x5 + 9x3 +14x1 = 65 + 27 + 14 = 106
> 
> The right side (UTE) has a weight of 106 too :
> 
> Ux1 + Tx3 +Ex5 = 21x1 + 20x3 + 5x5 = 21 + 60 + 25 = 106
> 
> So MINUTE is a « balanced word »
> 
> Some « balanced words » have an odd number of letters -- but
> the method is the same (I'll use a palindrome here):
> 
>                    R . A . D . A . R
>                    4 3 2 1 0 1 2 3 4
>                    --------^--------
> 
> Now, what about « balanced numbers » (this term already exists
> in the OEIS, unfortunately), using the same criteria?
> 
> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, ... 101, ... 111, ... are
> balanced numbers, of course, because palindromic -- but 7256 is
> too:
>                     7 . 2 . 5 . 6
>                     3 2 1 0 1 2 3
>                     ------^------
> 
>                 7x3 + 2x1 = 5x1 + 6x3       (=23)
> 
> I think a seq of such non-palindromic « balanced numbers » would
> start like this (computed by hand):
> 
> 1030, 1140, 1250, 1360, 1470, 1580, 1690, 2031, 2060, 2141, 2170...
> 
> Could someone check and submit this to Neil?
> Best,
> É.
> 
> 

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