A329279 Number of distinct tilings of a 2n X 2n square with 1 x n polyominoes.

1, 9, 11, 19, 22, 33, 37, 51, 56, 73, 79, 99, 106, 129, 137, 163, 172, 201, 211, 243, 254, 289, 301, 339, 352, 393, 407, 451, 466, 513, 529, 579, 596, 649, 667, 723, 742, 801, 821, 883, 904, 969, 991, 1059, 1082, 1153, 1177, 1251, 1276, 1353, 1379, 1459, 1486, 1569, 1597, 1683, 1712, 1801, 1831

Offset: 1

Author

Jeff Bowermaster, Nov 11 2019

Keywords

• nonn

Comments

The positions of n X n subsquares greatly restricts which permutations are possible, simplifying finding solutions. a(n+1) - a(n) = A014682 (n+2), where A014682 is the Collatz function, except a(2)-a(1) = 8 and A014682(4) = 5.

Links

• Jeff Bowermaster, Illustration of a(1)..a(3)
• Jeff Bowermaster, Illustration of a(4) and a(5)
• Jeff Bowermaster, Illustration of a(6)
• Jeff Bowermaster, Illustration of a(7)
• Jeff Bowermaster, Illustration of a(8)

Crossrefs

Cf. A014682, A060312, A058331 (bisection).

Programs

PARI

a(n) = if(n==1,1,if(n%2,(n^2+3*n)/2+2,(n^2+4*n)/2+3))

Formula

For even n, a(n) = (n^2+4n)/2+3; for odd n, a(n) = (n^2+3n)/2+2 ; a(1) = 1.