A243645 Number of ways two L-tiles can be placed on an n X n square.

0, 0, 0, 1, 20, 87, 244, 545, 1056, 1855, 3032, 4689, 6940, 9911, 13740, 18577, 24584, 31935, 40816, 51425, 63972, 78679, 95780, 115521, 138160, 163967, 193224, 226225, 263276, 304695, 350812, 401969, 458520, 520831, 589280, 664257, 746164, 835415, 932436

Offset: 0

Author

Alois P. Heinz, Jun 08 2014

Keywords

• nonn
• easy

Comments

This sequence also represents the number of edges added to G so that it is complete, where G is a graph of (n-1)^2 nodes arranged in a rhombus and embedded in the hexagonal lattice. G begins with A045944(n-2) edges and a(n) edges are added to form a complete graph. - John Tyler Rascoe, Sep 24 2022

Examples

			a(3) = 1:
._____.
|_| |_|
| |___|
|___|_| .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Column k=2 of A243608.

Programs

Maple

a:= n-> `if`(n<2, 0, ((((n-4)*n-1)*n+18)*n-16)/2):
seq(a(n), n=0..50);

Mathematica

CoefficientList[Series[x^3 (x^3+3x^2-15x-1)/(x-1)^5,{x,0,40}],x] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,0,0,1,20,87,244},40] (* Harvey P. Dale, Mar 06 2016 *)

Formula

G.f.: x^3*(x^3+3*x^2-15*x-1) / (x-1)^5.

a(n) = (n^4-4*n^3-n^2+18*n-16)/2 for n>=2, a(n) = 0 for n<2.

a(n) = A083374(n-1) - A045944(n-2) for n>=2. - John Tyler Rascoe, Sep 24 2022