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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.

%I A384219 #17 May 30 2025 18:47:35
%S A384219 2,6,24,104,442,1870,7920,33552,142130,602070,2550408,10803704,
%T A384219 45765226,193864606,821223648,3478759200,14736260450,62423800998,
%U A384219 264431464440,1120149658760,4745030099482,20100270056686,85146110326224,360684711361584,1527884955772562
%N 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.
%C A384219 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.
%H A384219 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,4,1).
%F A384219 G.f.: 2*x*(x-1)/((x^2+1)*(x^2+4*x-1)). - _Alois P. Heinz_, May 22 2025
%F A384219 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
%e A384219 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.
%e A384219 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.
%e A384219 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.
%p A384219 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]:
%p A384219 seq(a(n), n=1..25);  # _Alois P. Heinz_, May 22 2025
%Y A384219 Cf. A000045, A014437.
%K A384219 nonn,easy
%O A384219 1,1
%A A384219 _Angela L. Brobson_, May 22 2025