A323847 -id:A323847 - cp's OEIS frontend

A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

0, 0, 0, 1, 0, 1, 3, 4, 4, 3, 6, 16, 25, 16, 6, 10, 41, 94, 94, 41, 10, 15, 85, 266, 386, 266, 85, 15, 21, 155, 632, 1247, 1247, 632, 155, 21, 28, 259, 1332, 3423, 4657, 3423, 1332, 259, 28, 36, 406, 2570, 8342, 14795, 14795, 8342, 2570, 406, 36, 45, 606, 4631, 18546, 41586, 54219, 41586, 18546, 4631, 606, 45

Offset: 1

N. J. A. Sloane, Feb 06 2019

The monotonicity condition requires that M_{(i+1),j} = M_{i,j} + (0 or 1); M_{i,(j+1)} = M_{i,j} + (0 or 1).
These matrices can be cut into three connected pieces, containing the 0's, 1's, and 2's; there are two vertex-disjoint paths from the north-and-east edges of the matrix to the south-and-west edges.
Row (or column) n >= 1 has a linear recurrence (with constant coefficients) of order 2n+1. - Alois P. Heinz, Feb 07 2019

			Array begins:
    0   0    1    3     6    10 ...
    0   0    4   16    41    85 ...
    1   4   25   94   266   632 ...
    3  16   94  386  1247  3423 ...
    6  41  266 1247  4657 14795 ...
   10  85  632 3427 14795 54219 ...
...
The 4 examples when m=2 and n=3 are
    011   011  012   012
    012   112  012   112

D. E. Knuth, Email to N. J. A. Sloane, Feb 05 2019.

Alois P. Heinz, Antidiagonals n = 1..80, flattened

Rows 1-10 give: A000217(n-2), A323847, A323967, A323968, A323969, A323970, A323971, A323972, A323973, A323974.
Main diagonal gives A306322.
Cf. A132823, A252876, A229428.

More terms from Alois P. Heinz, Feb 07 2019

A381555 Triangle read by rows T(n,k) is the number of diamond coverings for a specific number of diamonds covering an even length row of triangles.

Original entry on oeis.org

1, 4, 1, 8, 4, 1, 13, 16, 4, 1, 19, 41, 24, 4, 1, 26, 85, 85, 32, 4, 1, 34, 155, 231, 145, 40, 4, 1, 43, 259, 532, 489, 221, 48, 4, 1, 53, 406, 1092, 1365, 891, 313, 56, 4, 1, 64, 606, 2058, 3333, 2926, 1469, 421, 64, 4, 1, 76, 870, 3630, 7359, 8294, 5551, 2255, 545, 72, 4

Offset: 0

Craig Knecht, Feb 27 2025

The total number of ways the diamond can cover a single row of length(n) triangles is the Fibonacci series. This total can be subdivided into categories based on the number of covering diamonds. The number of categories increase with the length of the row providing the structure of the triangle (see illustrations in the link below).
The above process provides a way to subdivide the individual Fibonacci numbers.
Comparing the diamond covering of a row of triangles shown here to the diamond corona of a hexagon A380346 or a diamond A380416 may be instructive.
A381552 provides additional graphics that help explain the diamond covering.

			Triangle begins:
  1, 4;
  1, 8, 4;
  1, 13, 16, 4;
  1, 19, 41, 24, 4;
  1, 26, 85, 85, 32, 4;
  1, 34, 155, 231, 145, 40, 4;

Craig Knecht, Diamond covering of a even length row of triangles.
Craig Knecht, Geometric bisection of the Fibonacci sequence.
Craig Knecht, Subdividing the individual Fibonacci numbers.
Walter Trump, Diamond covering producing the Fibonacci sequence.

Cf. A063496, A081219, A102083, A323847, A380346, A380416, A381552.

cp's OEIS Frontend

A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

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