[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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