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A141385
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a(n) = 7*a(n-1) - 9*a(n-2) + a(n-3) with a(0)=3, a(1)=7, a(2)=31.
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2
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3, 7, 31, 157, 827, 4407, 23563, 126105, 675075, 3614143, 19349431, 103593805, 554625899, 2969386479, 15897666067, 85113810057, 455687062275, 2439682811479, 13061709929935, 69930511268509, 374397872321627
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OFFSET
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0,1
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COMMENTS
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The old definition given for this sequence was "A sequence obeying a third-order linear recurrence".
Ruling out finitely many exceptional terms, this sequence differs by a constant from several related enumerations with a slightly more complicated structure (fourth-order linear recurrence):
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LINKS
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G. P. Michon, Silent Prisms: A Screaming Game for Short-Sighted People.
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FORMULA
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G.f.: (3 - 14*x + 9*x^2)/(1 - 7*x + 9*x^2 - x^3).
a(n+3) = 7*a(n+2) - 9*a(n+1) + a(n).
a(n) = A^n + B^n + C^n, where, putting u = atan(sqrt(5319)/73), we have:
A = 5.3538557854308282... = (7 + 2*sqrt(22)*cos(u/3))/3,
B = 1.5235479602692093... = (7 - sqrt(22)*cos(u/3) + sqrt(66)*sin(u/3))/3,
C = 0.1225962542999624... = (7 - sqrt(22)*cos(u/3) - sqrt(66)*sin(u/3))/3.
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EXAMPLE
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a(0) = 3 = A^0 + B^0 + C^0, a(1) = 7 = A + B + C.
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MAPLE
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m:=30; S:=series( (3-14*x+9*x^2)/(1-7*x+9*x^2-x^3), x, m+1):
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MATHEMATICA
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LinearRecurrence[{7, -9, 1}, {3, 7, 31}, 40] (* Harvey P. Dale, May 25 2011 *)
CoefficientList[Series[(3 -14x +9x^2)/(1 -7x +9x^2 -x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 21 2012 *)
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PROG
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(Magma) I:=[3, 7, 31]; [n le 3 select I[n] else 7*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 21 2012
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (3-14*x+9*x^2)/(1-7*x+9*x^2-x^3) ).list()
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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