A353434 -id:A353434 - cp's OEIS frontend

A353433 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 0..m-1 such that no iterated difference has a common factor with m >= 1.

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 2, 0, 1, 1, 4, 0, 2, 0, 1, 1, 2, 12, 0, 2, 0, 1, 1, 6, 0, 28, 0, 2, 0, 1, 1, 4, 30, 0, 48, 0, 2, 0, 1, 1, 6, 0, 126, 0, 60, 0, 2, 0, 1, 1, 4, 18, 0, 444, 0, 60, 0, 2, 0, 1, 1, 10, 0, 54, 0, 1350, 0, 52, 0, 2, 0, 1

Offset: 0

Pontus von Brömssen, Apr 21 2022

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.

			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

Pontus von Brömssen, Plot of T(n,7) for 0 <= n <= 200

Cf. A353434, A353435.
Rows: A000012 (n=0), A000010 (n=1), A061780 (every second term of row n=2).
Columns: A000012 (m=1), A019590 (m=2), A040000 (m=3), A130706 (m=4 and m=6).

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.)

A353436 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 1..m-1 such that the determinant of the Hankel matrix of any odd number of consecutive terms is not divisible by m >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 0, 0, 1, 4, 9, 4, 0, 0, 1, 5, 16, 22, 4, 0, 0, 1, 6, 25, 48, 56, 0, 0, 0, 1, 7, 36, 104, 144, 114, 0, 0, 0, 1, 8, 49, 180, 444, 320, 240, 0, 0, 0, 1, 9, 64, 298, 900, 1566, 720, 376, 0, 0, 0, 1, 10, 81, 468, 1828, 3744, 5576, 1312, 584, 0, 0, 0

Offset: 0

Pontus von Brömssen, Apr 21 2022

			Array begins:
  n\m| 1  2  3   4    5      6       7        8         9
  ---+---------------------------------------------------
   0 | 1  1  1   1    1      1       1        1         1
   1 | 0  1  2   3    4      5       6        7         8
   2 | 0  1  4   9   16     25      36       49        64
   3 | 0  0  4  22   48    104     180      298       468
   4 | 0  0  4  56  144    444     900     1828      3444
   5 | 0  0  0 114  320   1566    3744     9812     23208
   6 | 0  0  0 240  720   5576   15552    52784    157104
   7 | 0  0  0 376 1312  16544   54216   249424    968616
   8 | 0  0  0 584 2400  49900  189468  1191264   5991624
   9 | 0  0  0 724 3232 124052  550728  4955824  33844176
  10 | 0  0  0 920 4560 314932 1604088 20623232 191898648

Cf. A353434, A353435.
Rows: A000012 (n=0), A001477 (n=1), A000290 (n=2).
Columns: A000007 (m=1), A130716 (m=2).

T(n,m) = A353435(n,m) if m is prime.
T(n,1) = 0 if n >= 1.
T(n,2) = 0 if n >= 3.
T(n,3) = 0 if n >= 5.
T(n,4) = 0 if n >= 25.
T(n,5) = 0 if n >= 23.

cp's OEIS Frontend

A353433 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 0..m-1 such that no iterated difference has a common factor with m >= 1.

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