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A110272 a(n) = Pell(n)^3. 6
0, 1, 8, 125, 1728, 24389, 343000, 4826809, 67917312, 955671625, 13447314152, 189218084021, 2662500456000, 37464224551181, 527161643971768, 7417727240640625, 104375343011770368, 1468672529408250769 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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
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.
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.
MATHEMATICA
Fibonacci[Range[0, 30], 2]^3 (* G. C. Greubel, Sep 17 2021 *)
PROG
(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
(Sage) [lucas_number1(n, 2, -1)^3 for n in (0..30)] # G. C. Greubel, Sep 17 2021
CROSSREFS
Sequence in context: A016791 A061103 A264143 * A215040 A033536 A355762
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Jul 18 2005
STATUS
approved

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Last modified April 22 14:38 EDT 2024. Contains 371905 sequences. (Running on oeis4.)