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A188048
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Expansion of (1 - x^2)/(1 - 3*x^2 - x^3).
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5
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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
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OFFSET
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0,3
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COMMENTS
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Sequence is related to rhombus substitution tilings. For the tridiagonal unit-primitive matrix
U_1=
(0 1 0 0)
(1 0 1 0)
(0 1 0 1)
(0 0 1 1),
let M=(m_(i,j))=(U_1)^n, i,j=1,2,3,4. Then a(n)=m_(3,3).
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LINKS
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Robert Israel, Table of n, a(n) for n = 0..3286
L. E. Jeffery, Unit-primitive matrix
Index entries for linear recurrences with constant coefficients, signature (0, 3, 1).
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FORMULA
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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.
Alternatively, 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).
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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A052931.
Sequence in context: A188652 A333958 A114852 * A191529 A095132 A028940
Adjacent sequences: A188045 A188046 A188047 * A188049 A188050 A188051
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KEYWORD
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nonn,easy
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AUTHOR
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L. Edson Jeffery, Mar 19 2011
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STATUS
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approved
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