A353433 -id:A353433 - cp's OEIS frontend

A353434 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 no iterated difference is divisible by m >= 1.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 6, 2, 0, 0, 1, 5, 12, 8, 2, 0, 0, 1, 6, 20, 28, 6, 2, 0, 0, 1, 7, 30, 64, 48, 6, 2, 0, 0, 1, 8, 42, 126, 164, 60, 6, 2, 0, 0, 1, 9, 56, 216, 444, 336, 60, 6, 2, 0, 0, 1, 10, 72, 344, 954, 1350, 552, 52, 6, 2, 0, 0

Offset: 0

Pontus von Brömssen, Apr 21 2022

			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 | 0  1  2  3  4    5     6      7       8        9
   2 | 0  0  2  6 12   20    30     42      56       72
   3 | 0  0  2  8 28   64   126    216     344      512
   4 | 0  0  2  6 48  164   444    954    1850     3240
   5 | 0  0  2  6 60  336  1350   3630    8732    18240
   6 | 0  0  2  6 60  552  3582  11898   36290    90624
   7 | 0  0  2  6 52  772  8550  33862  133628   398048
   8 | 0  0  2  6 48 1054 17364  83946  437666  1545468
   9 | 0  0  2  6 48 1614 30126 182134 1278314  5300824
  10 | 0  0  2  6 48 2740 44922 346638 3321680 16079024

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

Cf. A350529, A353433, A353436.
Rows: A000012 (n=0), A001477 (n=1), A002378 (n=2), A245996 (n=3).
Columns: A000007 (m=1), A019590 (m=2), A040000 (m=3).

T(n,m) = A353433(n,m) if m is prime.
T(n,1) = 0 for n >= 1.
T(n,2) = 0 for n >= 2.
T(n,3) = 2 for n >= 1.
T(n,4) = 6 for n >= 4.
T(n,5) = 48 for n >= 8.
It appears that T(n,7) = T(n+42,7) for n >= 56. (See linked plot.)

A353435 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 the Hankel matrix of any odd number of consecutive terms is invertible over the ring of integers modulo m >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 0, 1, 1, 4, 4, 4, 0, 1, 1, 2, 16, 0, 4, 0, 1, 1, 6, 4, 48, 0, 0, 0, 1, 1, 4, 36, 0, 144, 0, 0, 0, 1, 1, 6, 16, 180, 0, 320, 0, 0, 0, 1, 1, 4, 36, 0, 900, 0, 720, 0, 0, 0, 1, 1, 10, 16, 108, 0, 3744, 0, 1312, 0, 0, 0, 1

Offset: 0

Pontus von Brömssen, Apr 21 2022

T(n,m) is divisible by T(2,m) = A127473(n) for n >= 2, because if r and s are coprime to m, the sequence (x_1, ..., x_n) satisfies the conditions if and only if the sequence (r*s^0*x_1 mod m, ..., r*s^(n-1)*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  1  4  4   16  4       36 16  36 16
   3 | 1  0  4  0   48  0      180  0 108  0
   4 | 1  0  4  0  144  0      900  0 324  0
   5 | 1  0  0  0  320  0     3744  0   0  0
   6 | 1  0  0  0  720  0    15552  0   0  0
   7 | 1  0  0  0 1312  0    54216  0   0  0
   8 | 1  0  0  0 2400  0   189468  0   0  0
   9 | 1  0  0  0 3232  0   550728  0   0  0
  10 | 1  0  0  0 4560  0  1604088  0   0  0
  11 | 1  0  0  0 4656  0  3895560  0   0  0
  12 | 1  0  0  0 4928  0  9467856  0   0  0
  13 | 1  0  0  0 4368  0 19185516  0   0  0

Cf. A035287, A045991, A350364, A353433, A353436.
Rows: A000012 (n=0), A000010 (n=1), A127473 (n=2).
Columns: A000012 (m=1), A130716 (m=2), A166926 (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) = A353436(n,m) if m is prime.
T(3,m) = (m-1)^2*(m-2) = A045991(m-1) if m is prime.
T(4,m) = (m-1)^2*(m-2)^2 = A035287(m-1) if m is prime.
Empirically: T(5,m) = (m-1)^2*(m-3)*(m^2-4*m+5) if m >= 3 is prime.
T(n,2) = 0 for n >= 3.
T(n,3) = 0 for n >= 5.
T(n,5) = 0 for n >= 23.

cp's OEIS Frontend

A353434 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 no iterated difference is divisible by m >= 1.

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