[seqfan] Re: A123712 and A178212

Mats Granvik mgranvik at abo.fi
Sat Feb 11 14:30:18 CET 2012


The previous message on this subject:
http://list.seqfan.eu/pipermail/seqfan/2012-February/016394.html

I did not manage to post the seqcomp list so I post here.

The first column in triangle:
https://oeis.org/A123706
is:
1,-2,-1,1,-1,2,-1,0,0,2,-1,-1,-1,2,1,0,-1,0...

By looking it up we get sequence:
https://oeis.org/A092673
which has the definition:
a(n) = moebius(n)- moebius(n/2) where moebius(n) is zero if n is not  
an integer.

By replacing:
a(n) = moebius(n)- moebius(n/2)
with:
a(n) = moebius(n/1)- moebius(n/2)

and knowing that moebius(n/1) is the first column in triangle:
https://oeis.org/A054525
And that moebius(n/2) (where moebius(n) is zero if n is not an integer),
is the second column in the same triangle A054525,

we can guess that triangle:
https://oeis.org/A123706
can be defined in terms of the Moebius function as:
A123706(n,k) = moebius(n/k)- moebius(n/(k+1)) where moebius(n) is zero  
if n is not an integer.

Or backwards:
A054525(n,k)= Sum from k = Infinity to k = b of A123706(n,k), where b  
< Infinity.

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

I don't know what proofs are but here is an observation I would
use as a starting point:

Consider the 3*3 matrix "A" of random numbers:
A(n,k)=
5,3,7
2,3,4
3,2,51

Which has the matrix inverse:
B(n,k)=
0.345238095,	-0.330952381	-0.021428571
-0.214285714,	0.557142857	-0.014285714
-0.011904762,	-0.002380952	0.021428571

Partial sums give:
C(n,k)=Sum from n=1 to n=3 of B(n,k)
C(n,k)=
0.345238095	-0.330952381	-0.021428571
0.130952381	0.226190476	-0.035714286
0.119047619	0.223809524	-0.014285714

Also consider the first differences of "A" as matrix "D":
D(n,k)=A(n,k)-A(n,k+1)
D(n,k)=
5-3, 3-7, 7-0
2-3, 3-4, 4-0
3-2, 2-51, 51-0
=
2,-4,7
-1,-1,4
1,-49,51

Which has the matrix inverse
E(n,k)=
0.345238095	-0.330952381	-0.021428571
0.130952381	0.226190476	-0.035714286
0.119047619	0.223809524	-0.014285714

To conclude: It appears that "C" equals "E" or
put longer: The matrix inverse "E" of the first
differences "D" of a matrix "A" equals the partials
sums "C" of a matrix "B" which is the
matrix inverse of matrix "A".

Mats





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