A258643 -id:A258643 - 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).

A261242 Irregular triangle T(n, k) of number of connected bisymmetric n X n matrices B_n with 0 or 1 entries, B_n[1,1] = 1 = B_n[1,n], and k islands of 0's.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 12, 18, 12, 8, 6, 2, 44, 56, 120, 28, 88, 4, 36, 0, 8

Offset: 1

Wolfdieter Lang and Kival Ngaokrajang, Aug 18 2015

The row length sequence is 1 for n = 1 and A000982(n-2) + 1 for n >= 2, that is:  1, 1, 2, 3, 6, 9, 14, 19, 26, 33, 42, ... = A261243.
This entry is motivated by A258643.
For bisymmetric matrices see the Wikipedia link.
For the number of independent entries of an n X n bisymmetric matrix B_n see a Jul 07 2015 comment on A002620(n+1), n >= 1. For the binary case (only 0 and 1 entries) see A060656(n+1), and the Dennis P. Walsh comment and link. If B_n[1,1] and B_n[1,n] is given then the four corners are fixed, and, for n >= 3, there are A002620(n+1) - 2 = A014616(n-2) entries free.
If the n X n bisymmetric matrix B_n of 0's and 1's with B_n[1, 1] = 1 = B_n[1, n] is considered as a grid of n^2 squares of length 1 (in some length unit) with the four corners filled with 1's and the other squares with 0 or 1 then a path between the centers of squares with step length 1 can be defined. No diagonal steps (length sqrt(2)) are allowed. B_n is called connected if there exists no path of 0's which dissects the grid into two parts.
An island of 0's (a 0-island) in B_n is defined as a set of 0's for which each pair is connected by a path of 0's, and a 0 entry at the coast of a 0-island has at least one entry 1 one step away. A single square filled with a 0 is a 0-island if all four neighbors 1 step (of length 1) apart are filled with 1's. If k=0 there exists no such 0-island. See the n=4 examples with k >=1 below. The k = 1 matrix has one simply connected 0-island of four squares. The four k = 2 matrices have two 0-islands consisting of one square each.
See the link with the figures by K. N. where red squares stand for 1 and empty squares for 0. Each matrix appears there rotated by 45 degrees in the counterclockwise direction. The mirror operation means row reversion in the matrix B_n. In the figures this is a mirror operation w.r.t. the middle NW-SE diagonal. 0-islands appear in the figures as holes.
For the row sums see A261244.

			The irregular triangle T(n, k) begins:
n\k   0   1    2   3   4  5   6   7   8  ...
1:    1
2:    1
3:    2   1
4:    4   1    4
5:   12  18   12   8   6  2
6:   44  56  120  28  88  4  36   0   8
...
n=4: k=0:
[[1,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]],
[[1,0,0,1], [0,1,1,0], [0,1,1,0], [1,0,0,1]],
[[1,1,0,1], [1,1,1,0], [0,1,1,1], [1,0,1,1]],
[[1,0,1,1], [0,1,1,1], [1,1,1,0], [1,1,0,1]];
     k=1:
[[1,1,1,1], [1,0,0,1], [1,0,0,1], [1,1,1,1]];
     k=2:
[[1,1,1,1], [1,0,1,1], [1,1,0,1], [1,1,1,1]],
[[1,1,1,1], [1,1,0,1], [1,0,1,1], [1,1,1,1]],
[[1,1,0,1], [1,0,1,0], [0,1,0,1], [1,0,1,1]],
[[1,0,1,1], [0,1,0,1], [1,0,1,0], [1,1,0,1]].

Kival Ngaokrajang, Illustration of T(n,k) for n = 1..5, k >= 0, T(6,0), T(6,1), T(6,2), T(6,4), T(6,k) for k = 3, 5, 6, 8
Wikipedia, Bisymmetric Matrix.

Cf. A000982, A002620, A014616, A258643, A261243, A261244.

A263677 Irregular table read by rows, n >= 1, k >= 0: T(n,k) is the number of binary symmetric n X n matrices with k holes that are squares, where the axes of symmetry are horizontal and vertical.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 28, 15, 11, 0, 6, 3, 30, 16, 0, 0, 3, 3

Offset: 1

Kival Ngaokrajang, Oct 23 2015

Inspired by A258643 but the axes of symmetry are the same as in A262742. The mirror patterns and nonsquare holes are excluded.

			Irregular table begins:
n\k  0  1  2  3  4  5 ...
1    1
2    1
3    3  1
4    3  1
5   28 15 11  0  6  3
6   30 16  0  0  3  3
...

Kival Ngaokrajang, Illustration of initial terms

Cf. A258643, A262742.

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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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A261242 Irregular triangle T(n, k) of number of connected bisymmetric n X n matrices B_n with 0 or 1 entries, B_n[1,1] = 1 = B_n[1,n], and k islands of 0's.

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A263677 Irregular table read by rows, n >= 1, k >= 0: T(n,k) is the number of binary symmetric n X n matrices with k holes that are squares, where the axes of symmetry are horizontal and vertical.

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