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A053524 (6^n - (-2)^n)/8. 6
0, 1, 4, 28, 160, 976, 5824, 35008, 209920, 1259776, 7558144, 45349888, 272097280, 1632587776, 9795518464, 58773127168, 352638730240, 2115832446976, 12694994550784, 76169967566848, 457019804876800, 2742118830309376, 16452712979759104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,12).

FORMULA

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

a(n) = (3^n-(-1)^n) * 2^n/8.

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

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

MAPLE

A053524:=n->(6^n-(-2)^n)/8; seq(A053524(n), n=0..40); # Wesley Ivan Hurt, Mar 11 2014

MATHEMATICA

Table[(6^n - (-2)^n)/8, {n, 0, 40}] (* Vincenzo Librandi, Mar 11 2014 *)

PROG

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

(MAGMA) [2^n/8*(3^n-(-1)^n): n in [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^100)) \\ Colin Barker, Mar 11 2014

CROSSREFS

Cf. A015518.

Sequence in context: A128721 A273647 A272936 * A270943 A272281 A182190

Adjacent sequences:  A053521 A053522 A053523 * A053525 A053526 A053527

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified October 22 01:42 EDT 2018. Contains 316431 sequences. (Running on oeis4.)