A220363 a(n) = Fibonacci(n)^3 + (-1)^n*Fibonacci(n+2).

1, -1, 4, 3, 35, 112, 533, 2163, 9316, 39215, 166519, 704736, 2986361, 12648727, 53583620, 226979403, 961507387, 4072998992, 17253519469, 73087050795, 309601764836, 1311494041879, 5555578042799, 23533806034368, 99690802469425, 422297015444207

Offset: 0

Author

Michel Marcus, Dec 12 2012

Keywords

• sign
• easy

Comments

An integral pentagon is a pentagon with integer sides and diagonals. There are two types of such pentagons.

Type B have sides A056570(n+2), A056570(n+2), a(n+2), A056570(n+2), A056570(n+2), and opposite diagonals A220362(n+2), A066258(n+2), A066258(n+2), A066258(n+2), A220362(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 = 0..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).

Programs

Mathematica

Table[Fibonacci[n]^3 + (-1)^n * Fibonacci[n + 2], {n, 0, 30}] (* T. D. Noe, Dec 13 2012 *)
LinearRecurrence[{3,6,-3,-1},{1,-1,4,3},30] (* Harvey P. Dale, Mar 19 2022 *)

PARI

Vec((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

Formula

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