A261242 -id:A261242 - cp's OEIS frontend

A261243 Row lengths of the irregular triangles A258643 and A261242: maximal number of 0-islands (holes) of certain bisymmetric n X n matrices with 0 or 1 entries only.

1, 1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926

Offset: 1

Wolfdieter Lang, Aug 18 2015

A shifted version of A061925. - R. J. Mathar, Aug 23 2015

Cf. A000982, A258643, A261242.

a(n) = ceiling(((n-2)^2)/2) + 1, n >= 2, a(1) = 1.
a(n) = (1/2)*(n-2)^2+1 if n is even,  a(n) = (ceiling((n-2)/2))^2 + (floor((n-2)/2))^2  + 1 if n is odd >= 3, and a(1) = 1.
O.g.f.: x*(1 - x + x^3 + x^4)/((1-x^2)*(1-x)^2) (from the o.g.f. of A000982).

A261244 Row sums of A261242: number of connected binary bisymmetric n X n matrices B with B[1,1] = 1 = B[2,2].

Original entry on oeis.org

1, 1, 3, 9, 58, 384

Offset: 1

Wolfdieter Lang and Kival Ngaokrajang, Aug 18 2015

For the definition of a connected binary bisymmetric n X n matrix B with the four corners occupied by 1s see the comment under A261242.

Cf. A261242.

A258643 Irregular triangle read by rows, n >= 1, k >= 0: T(n,k) is the number of distinct patterns of n X n squares with k holes that are squares (see the construction rule in comments).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 9, 7, 4, 4, 5, 2, 25, 11, 40, 8, 33, 3, 16, 0, 4

Offset: 1

Kival Ngaokrajang, Jun 06 2015

The sequence of row lengths is A261243. - Wolfdieter Lang, Aug 18 2015
The construction rules are: (o) The n X n square has horizontal and vertical diagonals. (i) A pattern must be symmetric with respect to both vertical and horizontal axes. (ii) For n >= 2, each pattern must have four squares at the corners. (iii) The squares must have continuity contact to each other either by sides or corners. (iv) The hole(s) must be square(s). Mirror patterns with respect to the main diagonal are not considered as different. See illustration in the links.
Each pattern can be a seed of a box fractal; e.g., the second pattern of T(3,0), consisting of 5 squares and 0 holes, is a seed of the Vicsek fractal (see a link below); the second pattern of T(4,2), consisting of 10 squares and 2 holes, is a seed of the fractal in a link of A002276.
If the figures are rotated by 45 degrees in the clockwise direction they can be considered as binary bisymmetric n X n matrices B_n if a red square stand for 1 and an empty square for 0. The four corners have entries 1, that is B_n[1, 1] = 1 = B_n[1, n]. The continuity of the red squares, mentioned above in point (iii), means that there is no rectangular path of 0's (no diagonal steps) in the matrix B_n that dissects it into two parts. See A261242 for more details, where also the figures with nonsquare holes and the mirrors (row reversion in the B_n matrix) are considered. - Wolfdieter Lang, Aug 18 2015

			Irregular triangle begins:
n\k  0   1   2  3   4  5   6  7  8 ...
1    1
2    1
3    2   1
4    3   1   2
5    9   7   4  4   5  2
6   25  11  40  8  33  3  16  0  4
...

Kival Ngaokrajang, Illustration of pattern T(n,k) for n = 1..5, k >= 0, T(6,k) patterns
Wikipedia, Vicsek fractal

Cf. A002276 (10 squares, 2 holes), A016203 (8 squares, 0 holes), A023001 (8 squares, 1 hole), A218724 (21 squares, 4 holes).

Cf. A261242, A261243.

A262666 Irregular table read by rows: T(n,k) is the number of binary bisymmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 4, 0, 8, 0, 12, 0, 14, 0, 12, 0, 8, 0, 4, 0, 1, 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1, 1, 0, 6, 0, 21, 0, 56, 0, 120, 0, 216, 0, 336, 0, 456, 0

Offset: 0

Kival Ngaokrajang, Sep 26 2015

T(n,k) = 0 if n is even and k is odd.

T(n,k) = T(n,k+1) if n is odd and k is even.

			Irregular table begins:
n\k 0   1   2   3   4   5   6   7   8   9   ...
0:  1
1:  1   1
2:  1   0   2   0   1
3:  1   1   2   2   2   2   2   2   1   1
4:  1   0   4   0   8   0  12   0  14   0   ...
5:  1   1   4   4  10  10  20  20  31  31   ...
...

Alois P. Heinz, Rows n = 0..32, flattened
Kival Ngaokrajang, Illustration of initial terms
Dennis P. Walsh, Notes on binary bisymmetric matrices
Wikipedia, Bisymmetric Matrix

Row sums give A060656(n+1).
Columns k=0-3 give: A000012, A000035, A052928, A237420(n+1).
Cf. A184948, A261242.

T:= n-> seq(coeff((t->(1+x^2)^(n-t)*(1+x)^t*(1+x^4)^
      (((n-2)*n+t)/4))(irem(n, 2)), x, i), i=0..n^2):
seq(T(n), n=0..6);  # Alois P. Heinz, Sep 27 2015

G.f. for row n: (1+x)^t*(1+x^2)^(n-t)*(1+x^4)^(((n-2)*n+t)/4) where t = n mod 2. - Alois P. Heinz, Sep 27 2015

More terms from Alois P. Heinz, Sep 27 2015

A380392 Irregular triangle read by rows: T(n,k) is the number of n X n binary matrices containing k South-East paths of 1's connecting the top left and bottom right corners.

Original entry on oeis.org

1, 1, 1, 13, 2, 1, 461, 26, 13, 8, 1, 2, 1, 61708, 1454, 953, 568, 325, 112, 178, 76, 22, 46, 48, 2, 16, 4, 4, 8, 8, 0, 1, 2, 1, 32348492, 340768, 279142, 168300, 125121, 44436, 81857, 24666, 25375, 28182, 19759, 4476, 17477, 4334, 7123, 6436, 4314, 1708, 5534

Offset: 0

John Tyler Rascoe, Jan 23 2025

A South-East path of 1's in a binary matrix is a path of connected 1's with steps South (0,-1) and East (1,0). Here 1's are said to be connected if they are adjacent in the same row or column.

Conjecture: The average number of South-East paths of 1's in all n X n binary matrices is A001790(n-1)/A101926(n-1). - John Tyler Rascoe, Feb 21 2025

			Triangle begins:
     k=0   1   2  3  4  5  6
 n=0   1;
 n=1   1,  1;
 n=2  13,  2,  1;
 n=3 461, 26, 13, 8, 1, 2, 1;
 ...
For row n = 3 the possible South-East paths are:
 A.       B.       C.       D.       E.       F.
 [1 1 1]  [1 1 0]  [1 1 0]  [1 0 0]  [1 0 0]  [1 0 0]
 [0 0 1]  [0 1 1]  [0 1 0]  [1 1 1]  [1 1 0]  [1 0 0]
 [0 0 1]  [0 0 1]  [0 1 1]  [0 0 1]  [0 1 1]  [1 1 1]
The 3 X 3 matrix below does not contain any of the paths A-F so it is counted under T(3,0) = 461.
 [1 0 1]
 [1 1 1]
 [1 0 0]
The 3 X 3 matrix below contains paths A, B, and D so it is counted under T(3,3) = 8.
 [1 1 1]
 [1 1 1]
 [1 0 1]

John Tyler Rascoe, Rows n = 0..5, flattened
John Tyler Rascoe, Python program.

Cf. A000984 (row lengths), A001790, A002416 (row sums), A086266, A101926, A261242, A369285.

Python
```
# see links
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A261243 Row lengths of the irregular triangles A258643 and A261242: maximal number of 0-islands (holes) of certain bisymmetric n X n matrices with 0 or 1 entries only.

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A261244 Row sums of A261242: number of connected binary bisymmetric n X n matrices B with B[1,1] = 1 = B[2,2].

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A258643 Irregular triangle read by rows, n >= 1, k >= 0: T(n,k) is the number of distinct patterns of n X n squares with k holes that are squares (see the construction rule in comments).

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A262666 Irregular table read by rows: T(n,k) is the number of binary bisymmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2.

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A380392 Irregular triangle read by rows: T(n,k) is the number of n X n binary matrices containing k South-East paths of 1's connecting the top left and bottom right corners.

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