A110272 a(n) = Pell(n)^3.

0, 1, 8, 125, 1728, 24389, 343000, 4826809, 67917312, 955671625, 13447314152, 189218084021, 2662500456000, 37464224551181, 527161643971768, 7417727240640625, 104375343011770368, 1468672529408250769

Offset: 0

Author

Paul Barry, Jul 18 2005

Keywords

• nonn
• easy

Comments

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/3,2/3)-fences, black third-squares (1/3 X 1 pieces, always placed so that the shorter sides are horizontal), and white third-squares. A (w,g)-fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/6,5/6)-fences, black (1/6,1/3)-fences, and white (1/6,1/3)-fences. - Michael A. Allen, Dec 29 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 0..850
• Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
• Toufik Mansour, A formula for the generating functions of powers of Horadam's sequence, Australas. J. Combin. 30 (2004) 207-212.
• Index entries for linear recurrences with constant coefficients, signature (12,30,-12,-1).

Crossrefs

Cf. A000129, A079291, A213688.

Programs

Magma

I:=[0,1,8,125]; [n le 4 select I[n] else 12*Self(n-1) + 30*Self(n-2) -12*Self(n-3) - Self(n-4): n in [1..31]]; // G. C. Greubel, Sep 17 2021

Mathematica

Fibonacci[Range[0, 30], 2]^3 (* G. C. Greubel, Sep 17 2021 *)

Sage

[lucas_number1(n, 2, -1)^3 for n in (0..30)] # G. C. Greubel, Sep 17 2021

Formula

G.f.: x*(1-4*x-x^2) / ((1+2*x-x^2)*(1-14*x-x^2)).

a(n) = 12*a(n-1) + 30*a(n-2) - 12*a(n-3) - a(n-4).

a(n) = (Pell(3*n) - 3*(-1)^n*Pell(n))/8.