%I #19 Mar 19 2022 17:08:43
%S 1,-1,4,3,35,112,533,2163,9316,39215,166519,704736,2986361,12648727,
%T 53583620,226979403,961507387,4072998992,17253519469,73087050795,
%U 309601764836,1311494041879,5555578042799,23533806034368,99690802469425,422297015444207
%N a(n) = Fibonacci(n)^3 + (-1)^n*Fibonacci(n+2).
%C An integral pentagon is a pentagon with integer sides and diagonals. There are two types of such pentagons.
%C 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,...
%D R. K. Guy, Unsolved Problems in Number Theory, D20.
%H Indranil Ghosh, <a href="/A220363/b220363.txt">Table of n, a(n) for n = 0..1593</a>
%H J. H. Jordan, B. E. Peterson, <a href="http://projecteuclid.org/euclid.rmjm/1181072619">Almost regular integer Fibonacci pentagons</a>, Rocky Mountain J. Math. Volume 23, Number 1 (1993), 243-247.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-3,-1).
%F 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
%t Table[Fibonacci[n]^3 + (-1)^n * Fibonacci[n + 2], {n, 0, 30}] (* _T. D. Noe_, Dec 13 2012 *)
%t LinearRecurrence[{3,6,-3,-1},{1,-1,4,3},30] (* _Harvey P. Dale_, Mar 19 2022 *)
%o (PARI) Vec((x^2-4*x+1)/((x^2-x-1)*(x^2+4*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 23 2014
%o (PARI) a(n) = fibonacci(n)^3 + (-1)^n*fibonacci(n+2) \\ _Charles R Greathouse IV_, Feb 14 2017
%K sign,easy
%O 0,3
%A _Michel Marcus_, Dec 12 2012