A220361 a(n) = Fibonacci(n)^3 + (-1)^n*Fibonacci(n-2).

1, 7, 28, 123, 515, 2192, 9269, 39291, 166396, 704935, 2986039, 12649248, 53582777, 226980767, 961505180, 4073002563, 17253513691, 73087060144, 309601749709, 1311494066355, 5555578003196, 23533806098447, 99690802365743, 422297015611968, 1788878864731825

Offset

2,2

Comments

An integral pentagon is a pentagon with integer sides and diagonals. There are two types of such pentagons. Type A have sides A066259(n+1), A220360(n+1), A066259(n+1), A220360(n+1), A066259(n+1), and opposite diagonals A056570(n+2), A056570(n+2), A220361(n+2), A056570(n+2), A056570(n+2), for n=1,2,...

References

R. K. Guy, Unsolved Problems in Number Theory, D20.

Links

Indranil Ghosh, Table of n, a(n) for n = 2..1593

J. H. Jordan, B. E. Peterson, Almost regular integer Fibonacci pentagons, Rocky Mountain J. Math. Volume 23, Number 1 (1993), 243-247.

Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Formula

a(n) = 3*a(n-1)+6*a(n-2)-3*a(n-3)-a(n-4). G.f.: x^2*(x^2+4*x+1) / ((x^2-x-1)*(x^2+4*x-1)). - Colin Barker, Sep 23 2014

Maple

with(combinat): A220361:=n->fibonacci(n)^3+(-1)^n*fibonacci(n-2): seq(A220361(n), n=2..30); # Wesley Ivan Hurt, Apr 26 2017

Mathematica

Table[Fibonacci[n]^3 + (-1)^n * Fibonacci[n - 2], {n, 2, 30}] (* T. D. Noe, Dec 13 2012 *)
LinearRecurrence[{3, 6, -3, -1}, {1, 7, 28, 123}, 30] (* Harvey P. Dale, Jul 13 2021 *)

Prog

(PARI) Vec(x^2*(x^2+4*x+1)/((x^2-x-1)*(x^2+4*x-1)) + O(x^100)) \\ Colin Barker, Sep 23 2014
(PARI) a(n) = fibonacci(n)^3 + (-1)^n*fibonacci(n-2) \\ Charles R Greathouse IV, Feb 14 2017

Crossrefs

Sequence in context: A290913 A303406 A054626 * A219737 A316106 A359203

Adjacent sequences: A220358 A220359 A220360 * A220362 A220363 A220364

Keyword

nonn,easy

Author

Michel Marcus, Dec 12 2012

Status

approved