A188048 Expansion of (1 - x^2)/(1 - 3*x^2 - x^3).

1, 0, 2, 1, 6, 5, 19, 21, 62, 82, 207, 308, 703, 1131, 2417, 4096, 8382, 14705, 29242, 52497, 102431, 186733, 359790, 662630, 1266103, 2347680, 4460939, 8309143, 15730497, 29388368, 55500634, 103895601, 195890270, 367187437, 691566411, 1297452581

Offset

0,3

Comments

Sequence is related to rhombus substitution tilings.

Links

Robert Israel, Table of n, a(n) for n = 0..3286

L. Edson Jeffery, Unit-primitive matrix

Index entries for linear recurrences with constant coefficients, signature (0,3,1).

Formula

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

a(n) = 3*a(n-2)+a(n-3), for n>=3, with a(0)=1, a(1)=0, a(2)=2.

a(n) = a(n-1)+3*a(n-2)-2*a(n-3)-a(n-4), for n>=4, with {a(k)}={1,0,2,1}, k=0,1,2,3.

a(n) = A187497(3*n+1).

a(n) = m_(3,3), where (m_(i,j)) = (U_1)^n, i,j=1,2,3,4 and U_1 is the tridiagonal unit-primitive matrix [0, 1, 0, 0; 1, 0, 1, 0; 0, 1, 0, 1; 0, 0, 1, 1].

3*(-1)^n*a(n) = A215664(n). - Roman Witula, Aug 20 2012

a(2n) = A094831(n); a(2n+1) = A094834(n). - John Blythe Dobson, Jun 20 2015

a(n) = A052931(n)-A052931(n-2). - R. J. Mathar, Nov 03 2020

a(n) = (2^n/3)*(cos^n(Pi/9) + cos^n(5*Pi/9) + cos^n(7*Pi/9)). - Greg Dresden, Sep 24 2022

Maple

F:= gfun:-rectoproc({a(n)=3*a(n-2)+a(n-3), a(0)=1, a(1)=0, a(2)=2}, a(n), remember):
map(F, [$0..100]); # Robert Israel, Jun 21 2015

Mathematica

CoefficientList[Series[(1-x^2)/(1-3x^2-x^3), {x, 0, 40}], x]  (* Harvey P. Dale, Mar 31 2011 *)
LinearRecurrence[{0, 3, 1}, {1, 0, 2}, 50] (* Roman Witula, Aug 20 2012 *)

Prog

(PARI) abs(polsym(1-3*x+x^3, 66)/3) /* Joerg Arndt, Aug 19 2012 */
(Magma) I:=[1, 0, 2, 1]; [n le 4 select I[n] else Self(n-1)+3*Self(n-2)-2*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2015

Crossrefs

Cf. A052931.

Sequence in context: A188652 A333958 A114852 * A191529 A095132 A028940

Adjacent sequences: A188045 A188046 A188047 * A188049 A188050 A188051

Keyword

nonn,easy

Author

L. Edson Jeffery, Mar 19 2011

Status

approved