EricD./0and1pattern

[Eric]

yes,
i looked just at pairs of consecutive integers.
and i find this always interesting,
because
as Maximilian Hasler said

lines and columns :
2*3=6/2+6/3=5
3*4=12/2+12/3+12/4=13
4*5=20/2+20/3+20/4+20/5=25,6666667 THIS is interesting
...

on one hand i have (n^2+n^3) A011379
on other hand i have some irrationnal

what you say and your sequence is interesting thanks to take time to read
me,
and explain me some protocol.

i'm just learning whats is interesting to say
and interesting to mute

Best Regards
Eric


-----Message d'origine-----
De : franktaw at netscape.net [mailto:franktaw at netscape.net]
Envoye : lundi 24 mars 2008 21:11
A : seqfan at ext.jussieu.fr
Objet : Re: EricD./0and1pattern


First of all, the lengths are *not* the pronic numbers; instead they
are A003418 --
the LCM of (1,2,3,...,n).  (You would get the pronic numbers if you
looked just at
pairs of consecutive integers.  But this is not interesting; the number
of 1's is just
the odd numbers.)

Next, you are arbitrarily excluding the case k = 1:

1=1111111111111...

If you include it and get the length right, the number of 1's is
A025529.

Franklin T. Adams-Watters

P.S. don't call this the "sum" - sum means you are adding the numbers.
This is
a "concatenation" -- "vertical concatenation" would make it clear what
you mean.
(Although if you do add the numbers, you get the same length pattern,
and
the sum of the numbers in the pattern is the number of 1's in your
pattern: e.g.,
for n = 4 the sum is:

122313132214,

(i.e., 1,2,2,3,1,3,1,3,2,2,1,4 -- use ",", not "|", as a separator)

which sums to 25, the same as the number of 1's in

111111111111
010101010101
001001001001
000100010001.

I have submitted sequence A138553 -- titled "Table, T(n,k) is the
number of
divisors of n that are <= k". with these summed patterns)

-----Original Message-----
From: Eric <moongerms at wanadoo.fr>

Hello

first i consider :
2=01010101010101010101...
3=001001001001001001001001001001...
4=00010001000100010001000100010001...


i make sum of 2 and 3 and i have the pattern
010101
001001

sum of 2,3,4 and i have the pattern
010101010101
001001001001
000100010001

each time i have a pattern of a "n" numbers of columns (as width)
and as length (6,12,30,42,56,...) A002378  Oblong (or pronic, or
heteromecic) numbers: n(n+1)

so n*n*(n+1)=n^3+n^2 which is A011379  n^2+n^3.

for each pattern i consider the number of 0 and the number of 1,
number of 0 | 7|23|55|107|194
number of 1 | 5|13|25|43|58
with A011379 as tool i can go faster

i make a search for 7|23|55|107|194...i have 6821 results
i make a search for 5|13|25|43|58...i have 8528 results  lol :o)
...i'm very proud of my search lol   (joke)...

but for 7,23,55,107,194 and 5,13,25,43,58 i found nothing in database
i try with subtraction of each term (23-7,55-23,etc...),
there are some results but i dont see all of them in detail presently...

however,
for each pattern, more than the number of zero, i will see if there is
some
scheme (only for eyes pleasure without conviction)

And
we can take away a crescent quantity of 0,
by subtract two times A000217   Triangular numbers: a(n) = C(n+1,2) =
n(n+1)/2 = 0+1+2+...+n. for each pattern this is the first "triangular
hat"
of 0...and second "triangular hat" of 0

and for the 1 numbers i can subtract n*1 of each pattern

it give
number of 0 | 7|23|55|107|194
minus     3| 6|10| 15| 21
           4|17|45| 92|173 i have 2 results in database and this is NOT
A095667
minus     3|6 |10| 15| 21
          1|11|55| 77|152  i find nothing in database

number of 1 | 5|13|25|43|58
minus     2| 3| 4| 5| 6
          3|10|21|38|52 i find nothing in database

that's all for this mail
it was just to play with 1 and 0...
i hope my step is not tasteless
and that i dont fall in a trap of trivial properties of numbers.
(i fear for one's bacon lol)

PS : using the "|" caracter to separate numbers is not very efficient
but i
try it to test, and i'm surprised to see that the n integer sequence
(1,2,3,4,5,6...) doesn't appear first in all cases...
In spite of big number of result by using "|", perhaps the variation of
this
number could give some sign...like the space between each integer
inside one
sequence.


BestRegards
Eric