A144056 Number of n X n binary arrays, symmetric about both diagonal and antidiagonal, with every 1 adjacent to at least one other 1, but at most one 1 adjacent horizontally and at most one 1 adjacent vertically.

1, 2, 3, 11, 31, 171, 719, 5907, 40576, 574191, 6118805, 134458817, 2328856652, 81657178991, 2239533631682, 126154274091928, 5530047112752039, 494646201440552586, 34629959352847199652, 4953721399562434015229