A053524 a(n) = (6^n - (-2)^n)/8.

0, 1, 4, 28, 160, 976, 5824, 35008, 209920, 1259776, 7558144, 45349888, 272097280, 1632587776, 9795518464, 58773127168, 352638730240, 2115832446976, 12694994550784, 76169967566848, 457019804876800, 2742118830309376, 16452712979759104

Offset: 0

Author

N. J. A. Sloane, Barry E. Williams, Jan 15 2000

Keywords

• nonn
• easy

Comments

The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.1(b).
• A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 194-196.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
• Index entries for linear recurrences with constant coefficients, signature (4,12).

Crossrefs

Cf. A015518.

Programs

Magma

[2^n/8*(3^n-(-1)^n): n in [0..30]]; // Vincenzo Librandi, Mar 11 2014

Maple

A053524:=n->(6^n-(-2)^n)/8; seq(A053524(n), n=0..30); # Wesley Ivan Hurt, Mar 11 2014

Mathematica

Table[(6^n -(-2)^n)/8, {n, 0, 30}] (* Vincenzo Librandi, Mar 11 2014 *)

PARI

a(n) = (6^n-(-2)^n)/8; \\ Joerg Arndt, Mar 11 2014

PARI

Vec(-x/((2*x+1)*(6*x-1)) + O(x^30)) \\ Colin Barker, Mar 11 2014

Sage

[lucas_number1(n,4,-12) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009

Formula

E.g.f.: (exp(6*x) - exp(-2*x))/8.

a(n) = 2^(n-3) * (3^n - (-1)^n) = 2^(n-3)*A105723(n).

a(n) = 4*a(n-1) + 12*a(n-2), with a(0)=0, a(1)=1.

G.f.: x / ((1+2*x)*(1-6*x)). - Colin Barker, Mar 11 2014