OFFSET
0,8
COMMENTS
T(n,m) is divisible by T(1,m) = A000010(m) if n >= 1, because if r is coprime to m, the sequence (x_1, ..., x_n) satisfies the conditions if and only if the sequence (r*x_1 mod m, ..., r*x_n mod m) does.
LINKS
Pontus von Brömssen, Plot of T(n,7) for 0 <= n <= 200
FORMULA
For fixed n, T(n,m) is multiplicative with T(n,p^e) = T(n,p)*p^(n*(e-1)).
T(n,m) = A353434(n,m) if m is prime.
For each n >= 0, there exists an n-th degree polynomial P such that T(n,m) = P(m) for sufficiently large primes m. For example (for n >= 4, these are empirical observations only):
T(0,m) = 1 for all m >= 1;
T(1,m) = m-1 for all primes m;
T(2,m) = (m-1)*(m-2) for all primes m;
T(3,m) = (m-1)*(m^2-5*m+7) for primes m >= 3;
T(4,m) = (m-1)*(m^3-9*m^2+30*m-38) for primes m >= 5;
T(5,m) = (m-1)*(m^4-14*m^3+81*m^2-235*m+302) for primes m >= 7;
T(6,m) = (m-1)*(m^5-20*m^4+175*m^3-854*m^2+2401*m-3280) for primes m >= 19.
T(n,2) = 0 for n >= 2.
T(n,3) = 2 for n >= 1.
T(n,5) = 48 for n >= 8.
It appears that T(n,7) = T(n+42,7) for n >= 56. (See linked plot.)
EXAMPLE
Array begins:
n\m| 1 2 3 4 5 6 7 8 9 10
---+------------------------------------
0 | 1 1 1 1 1 1 1 1 1 1
1 | 1 1 2 2 4 2 6 4 6 4
2 | 1 0 2 0 12 0 30 0 18 0
3 | 1 0 2 0 28 0 126 0 54 0
4 | 1 0 2 0 48 0 444 0 162 0
5 | 1 0 2 0 60 0 1350 0 486 0
6 | 1 0 2 0 60 0 3582 0 1458 0
7 | 1 0 2 0 52 0 8550 0 4374 0
8 | 1 0 2 0 48 0 17364 0 13122 0
9 | 1 0 2 0 48 0 30126 0 39366 0
10 | 1 0 2 0 48 0 44922 0 118098 0
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Pontus von Brömssen, Apr 21 2022
STATUS
approved