A099215 a(n) = 4*a(n-1) - 4*a(n-2) + 3*a(n-3).

1, 2, 4, 11, 34, 104, 313, 938, 2812, 8435, 25306, 75920, 227761, 683282, 2049844, 6149531, 18448594, 55345784, 166037353, 498112058, 1494336172, 4483008515, 13449025546, 40347076640, 121041229921, 363123689762, 1089371069284

Offset

0,2

Comments

Binomial transform of A122552. The alternating sums of 3 consecutive terms are powers of 3. - Klaus Purath, Dec 11 2025

Links

Table of n, a(n) for n=0..26.

Index entries for linear recurrences with constant coefficients, signature (4,-4,3).

Formula

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

a(n) = Sum_{k=0..floor(n/3)} binomial(n-k, 2*k)*3^k*2^(n-3*k).

For n>=1, a(n-1) = (1/7) * (-cos(Pi*n/3) + cosh(n*log(3)) + 3*sqrt(3)*sin(Pi*n/3) + sinh(n*log(3))). - Harvey P. Dale, Mar 02 2013

Mathematica

LinearRecurrence[{4, -4, 3}, {1, 2, 4}, 30] (* Harvey P. Dale, Mar 02 2013 *)

Crossrefs

Cf. A099214.

Sequence in context: A198634 A287007 A369843 * A089407 A289588 A362638

Adjacent sequences: A099212 A099213 A099214 * A099216 A099217 A099218

Keyword

easy,nonn

Author

Paul Barry, Oct 06 2004

Status

approved