A348134 Number of ways two L-tiles (with rotation) can be placed on an n X n square.

0, 0, 22, 336, 1422, 3952, 8790, 16992, 29806, 48672, 75222, 111280, 158862, 220176, 297622, 393792, 511470, 653632, 823446, 1024272, 1259662, 1533360, 1849302, 2211616, 2624622, 3092832, 3620950, 4213872, 4876686, 5614672, 6433302, 7338240, 8335342, 9430656

Offset

1,3

Comments

All terms are even, because groups of ways, which are connected by 90 degrees rotation symmetry, are made up from 4 or 2 ways, so the number of ways will be some 4m+2n, and 4m+2n is even.

Links

Table of n, a(n) for n=1..34.

Nicolas Bělohoubek, Visualization of 3rd term

Nicolas Bělohoubek, 90° rotation groups for 3rd term

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 2*(n - 2)*(4*n^3 - 8*n^2 - 19*n + 32) for n > 1.

G.f.: 2*x^3*(11 + 113*x - 19*x^2 - 9*x^3)/(1 - x)^5. - Stefano Spezia, Oct 03 2021

Example

For a(1) and a(2) there are fewer squares on the main square then squares of the 2 L-tiles, so a(1) = a(2) = 0.

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 22, 336, 1422, 3952}, 40] (* Harvey P. Dale, Mar 04 2023 *)

Crossrefs

Cf. A242856, A243645.

Sequence in context: A021284 A019623 A021794 * A223812 A018090 A021274

Adjacent sequences: A348131 A348132 A348133 * A348135 A348136 A348137

Keyword

nonn,easy

Author

Nicolas Bělohoubek, Oct 02 2021

Status

approved