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URL: https://oeis.org/A188048

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A188048
Expansion of (1 - x^2)/(1 - 3*x^2 - x^3).
5
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.
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
KEYWORD
nonn,easy
AUTHOR
L. Edson Jeffery, Mar 19 2011
STATUS
approved