A334995 Twice the area of triangle with coordinates (Fn, Fn+k), (Fn+2k, Fn+3k) and (Fn+4k, Fn+5k), where Fn is the n-th Fibonacci number A000045.

1, 15, 256, 2835, 33275, 368640, 4121741, 45703035, 507456256, 5627634375, 62422224679, 692270530560, 7677591693929, 85145750881815, 944284326022400, 10472272829590635, 116139347801260099, 1288005089535959040, 14284196451517672789, 158414165892802771875, 1756840041348774377216

Offset

1,2

Links

Table of n, a(n) for n=1..21.

S. Edwards, Elementary problem B-1172, Fibonacci Quarterly 53 (2015), 273.

S. Edwards, Elementary problem B-1172. Area of Triangles with Generalized Fibonacci Number Coordinates, Fibonacci Quarterly 54 (2016), 273-276.

Virginia P. Johnson and Charles K. Cook, Areas of Triangles and Other Polygons with Vertices from Various Sequences, Fibonacci Quart. 55 (2017), no. 5, 86-95.

Formula

a(n) = F(n)^2*L(n)^3 if n is even, 5*F(n)^4*L(n) if n is odd, where F(n) is the n-th Fibonacci number A000045(n), and L(n) is the n-th Lucas number A000032(n).

Conjectures from Colin Barker, May 19 2020: (Start)

G.f.: x*(1 + x^2)*(1 + 5*x + 74*x^2 - 5*x^3 + x^4) / ((1 + 4*x - x^2)*(1 + x - x^2)*(1 - 4*x - x^2)*(1 - 11*x - x^2)).

a(n) = 10*a(n-1) + 31*a(n-2) - 190*a(n-3) - 236*a(n-4) + 190*a(n-5) + 31*a(n-6)- 10*a(n-7) - a(n-8) for n>8.

(End)

Prog

(PARI) F(n) = fibonacci(n);
L(n) = F(n+1)+F(n-1);
a(n) = if (n%2, F(n)^2*L(n)^3, 5*F(n)^4*L(n));

Crossrefs

Cf. A000032, A000045.

Cf. A289668, A289669.

Sequence in context: A015691 A145562 A027775 * A099273 A206230 A209263

Adjacent sequences: A334992 A334993 A334994 * A334996 A334997 A334998

Keyword

nonn

Author

Michel Marcus, May 19 2020

Status

approved