[seqfan] Connection between A000245 and A026016 (or A071725)?
Ali Sada
pemd70 at yahoo.com
Sun Jun 7 21:52:17 CEST 2020
Hi Everyone,
Arrange the numbers between 1 and n in a raw of size 2n such that:
a) 1 is always in the first spot;
b) k's position in the raw is <= 2k; and
c) k always comes before k+1
For example, below are the possibilities we get when n = 3. (I put “*” instead of a space for more clarity.)
1, 2, 3, *,* ,*
1, 2, *, 3, *, *
1, 2, *, *, 3, *
1, 2, *, *, *, 3
1, *, 2, 3, *, *
1, *, 2, *, 3, *
1, *, 2, *, *, 3
1, *, *, 2, 3, *
1, *, *, 2, *, 3
The possibilities I got were 1, 3, 9, 28, 90, 297, 1001, 3432, 11934 (A000245.)
Then, I filled the remaining spaces with the numbers between n+1 and 2n.
Using n = 3 as an example we get:
1, 2, 3, 4, 5, 6
1, 2, 4, 3, 5, 6
1, 2, 4, 5, 3, 6
1, 2, 4, 5, 6, 3
1, 4, 2, 3, 5, 6
1, 4, 2, 5, 3, 6
1, 4, 2, 5, 6, 3
1, 4, 5, 2, 3, 6
1, 4, 5, 2, 6, 3
For the numbers I tested (only 8,) the sum of the third column minus the sum of the second column was A026016(n-1).
In the example of n = 3 above, the sum of the third column is 31, and the sum of the second column is 28.
31-28 = 3, which is A026016(2).
3 is also the result of the subtraction of the fourth column minus the third column, and the sixth column minus the fifth column.
I would really appreciate it if you could tell me if this pattern will continue and why.
Best,
Ali
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