A081335 a(n) = (6^n + 2^n)/2.

1, 4, 20, 112, 656, 3904, 23360, 140032, 839936, 5039104, 30233600, 181399552, 1088393216, 6530351104, 39182090240, 235092508672, 1410554986496, 8463329787904, 50779978465280, 304679870267392, 1828079220555776

Offset

0,2

Comments

Binomial transform of A034478. 4th binomial transform of (1, 0, 4, 0, 16, 0, 64, ...).

Case k=4 of the family of recurrences a(n) = 2*k*a(n-1) - (k^2-4)*a(n-2), a(0)=1, a(1)=k.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (8,-12).

Formula

a(n) = 8*a(n-1) - 12*a(n-2), a(0)=1, a(1)=4.

G.f.: (1-4*x)/((1-2*x)*(1-6*x)).

E.g.f.: exp(4*x)*cosh(2*x).

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * 4^(n-k) = Sum_{k=0..n} binomial(n,k) * 4^(n-k/2) * (1+(-1)^k)/2. - Paul Barry, Nov 22 2003

a(n) = Sum_{k=0..n} 4^k*A098158(n,k). - Philippe Deléham, Dec 04 2006

Mathematica

LinearRecurrence[{8, -12}, {1, 4}, 30] (* Harvey P. Dale, May 03 2013 *)
CoefficientList[Series[(1-4x)/((1-2x)(1-6x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 08 2013 *)

Prog

(Magma) [(6^n+2^n)/2: n in [0..30]]; // Vincenzo Librandi, Aug 08 2013
(PARI) a(n)=(6^n+2^n)/2 \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [2^(n-1)*(3^n + 1) for n in (0..30)] # G. C. Greubel, Aug 02 2019
(GAP) List([0..30], n-> 2^(n-1)*(3^n + 1)); # G. C. Greubel, Aug 02 2019

Crossrefs

Cf. A081336.

Sequence in context: A014523 A153299 A239643 * A136783 A227726 A080609

Adjacent sequences: A081332 A081333 A081334 * A081336 A081337 A081338

Keyword

nonn,easy

Author

Paul Barry, Mar 18 2003

Status

approved