A187273 a(n) = (n/4)*3^(n/2)*((1+sqrt(3))^2+(-1)^n*(1-sqrt(3))^2).

0, 3, 12, 27, 72, 135, 324, 567, 1296, 2187, 4860, 8019, 17496, 28431, 61236, 98415, 209952, 334611, 708588, 1121931, 2361960, 3720087, 7794468, 12223143, 25509168, 39858075, 82904796, 129140163, 267846264, 416118303, 860934420, 1334448351, 2754990144, 4261625379, 8781531084, 13559717115, 27894275208

Offset

0,2

Links

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

R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. See Lemma 1.

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-9).

Formula

From Colin Barker, Jul 24 2013: (Start)

a(n) = 6*a(n-2) - 9*a(n-4).

G.f.: 3*x*(x+1)*(3*x+1) / (3*x^2-1)^2. (End)

a(2*n) = 4*n*3^n, a(2*n+1) = (2*n+1)*3^(n+1). - Andrew Howroyd, Mar 28 2016

Maple

See A187272.

Mathematica

LinearRecurrence[{0, 6, 0, -9}, {0, 3, 12, 27}, 40] (* Harvey P. Dale, Apr 21 2014 *)
CoefficientList[Series[3 x (x + 1) (3 x + 1)/(3 x^2 - 1)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 23 2014 *)

Prog

(PARI) for(n=0, 50, print1(round((n/4)*3^(n/2)*((1+sqrt(3))^2+(-1)^n*(1-sqrt(3))^2)), ", ")) \\ G. C. Greubel, Jul 08 2018
(Magma) [Round((n/4)*3^(n/2)*((1+Sqrt(3))^2+(-1)^n*(1-Sqrt(3))^2)): n in [0..50]]; // G. C. Greubel, Jul 08 2018
(Python)
def A187273(n): return n*3**(1+(n>>1)) if n&1 else (n<<1)*3**(n>>1) # Chai Wah Wu, Feb 19 2024

Crossrefs

Sequence in context: A018230 A058034 A294416 * A361847 A009259 A183503

Adjacent sequences: A187270 A187271 A187272 * A187274 A187275 A187276

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 07 2011

Status

approved