A384219 Areas of triangles whose three vertices are consecutive ordered pairs of consecutive odd Fibonacci numbers such that an ordered pair’s y-value is the next ordered pair’s x-value.

2, 6, 24, 104, 442, 1870, 7920, 33552, 142130, 602070, 2550408, 10803704, 45765226, 193864606, 821223648, 3478759200, 14736260450, 62423800998, 264431464440, 1120149658760, 4745030099482, 20100270056686, 85146110326224, 360684711361584, 1527884955772562

Offset: 1

Author

Angela L. Brobson, May 22 2025

Keywords

• nonn
• easy

Comments

As described above, the vertices of each triangle are made up exclusively of odd integers in both coordinates. The area of a parallelogram spanned by (x_1, y_1), (x_2, y_2), and (x_3, y_3) is given by abs(x_1*(y_2 - y_3) + x_2*(y_3 - y_1) + x_3*(y_1 - y_2)). This value is even since each y_i - y_j is a difference of odd integers and is thereby even. As the area of a triangle formed from three given vertices is half the area of the parallelogram spanned by those vertices, it follows that the area of each triangle formed according to the description is an integer.

Examples

			The first term is created by finding the area of the triangle formed by the ordered pairs (1,1), (1,3), and (3,5), which is 2.
The second term is created by finding the area of the triangle formed by the ordered pairs (1,3), (3,5), and (5,13), which is 6.
The third term is created by finding the area of the triangle formed by the ordered pairs (3,5), (5,13), and (13,21), which is 24.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,0,4,1).

Crossrefs

Cf. A000045, A014437.

Programs

Maple

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|4|0|4>>^n.<<0, 2, 6, 24>>)[1,1]:
seq(a(n), n=1..25);  # Alois P. Heinz, May 22 2025

Formula

G.f.: 2*x*(x-1)/((x^2+1)*(x^2+4*x-1)). - Alois P. Heinz, May 22 2025

E.g.f.: (exp(2*x)*(cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x)) - cos(x) + 3*sin(x))/5. - Stefano Spezia, May 25 2025