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

1,2

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

Andrew Howroyd, Table of n, a(n) for n = 1..10000

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)

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

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

G.f.: x*(1 + 8*x - 8*x^3 + 3*x^5)/((1 - x)^3*(1 + x)^2). - Andrew Howroyd, Nov 14 2025

Mathematica

A329279[n_] := If[n == 1, 1, (n^2 + (3 + #)*n)/2 + 2 + # & [Boole[EvenQ[n]]]];
Array[A329279, 60] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 9, 11, 19, 22, 33}, 60] (* Paolo Xausa, Nov 15 2025 *)

Prog

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

Crossrefs

Cf. A014682, A060312, A058331 (bisection).

Sequence in context: A263722 A299971 A090771 * A372242 A284295 A284294

Adjacent sequences: A329276 A329277 A329278 * A329280 A329281 A329282

Keyword

nonn,easy

Author

Jeff Bowermaster, Nov 11 2019

Status

approved