OFFSET
1,6
COMMENTS
When taking matrix powers of the reversed triangle it might be more interesting to start with the first term T(1,1) set to 0.
Three fundamental number theoretic sequences are found from this triangle. The first is the Dirichlet inverse of the Euler totient which are the entries themselves. The Greatest Common Divisor-Fourier transform described by Wolfgang Schramm gives the Möbius function times n =1, -2, -3, 0, ... = A055615, as follows:
1*cos(-2*k*Pi/n) = 1
1*cos(-2*k*Pi/n) -1*cos(-2*k*Pi/n) = -2
1*cos(-2*k*Pi/n) +1*cos(-2*k*Pi/n) -2*cos(-2*k*Pi/n) = -3
The two components in this GCD-Fourier triangle both sum to the sequence 1,0,0,0,0, ... A000007.
1 = 1
1 -1 = 0
1 +1 -2 = 0
...
cos(-2*k*Pi/n) = 1
cos(-2*k*Pi/n), cos(-2*k*Pi/n) = 0
cos(-2*k*Pi/n), cos(-2*k*Pi/n), cos(-2*k*Pi/n) = 0
...
This latter Fourier transform like triangle is also called the chaotic set by some authors.
The third arithmetic sequence is the von Mangoldt function that can be computed as sums with periods equal to rows in this triangle:
1
log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2))
log(3) = Sum_{n>=0} (1/(n+1) +1/(n+2) -2/(n+3))
log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2) +1/(n+3) -1/(n+4))
log(5) = Sum_{n>=0} (1/(n+1) +1/(n+2) +1/(n+3) +1/(n+4) -4/(n+5))
log(1) = Sum_{n>=0} (1/(n+1) -1/(n+2) -2/(n+3) -1/(n+4) +1/(n+5) +2/(n+6))
...
Also the matrix inverse of the reversal of this number triangle gives the all-ones sequence in the first column. Therefore this number triangle is a companion to A054524.
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
T(n,k) = A023900(gcd(n,k)) for n >= k.
MATHEMATICA
Clear[nn, t, n, k]; nn = 12; t[n_, 1] = 1; t[1, k_] = 1;
t[n_, k_] := t[n, k] = If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[t[k - i, n], {i, 1, n - 1}]]; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]]
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Mats Granvik, Nov 19 2013
STATUS
approved