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A141385
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A sequence obeying a third-order linear recurrence.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| 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):
For n>0, A141221(n)=a(n)-1. For n>2, A141384(n)=a(n)+1.
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LINKS
| G. P. Michon, Silent Prisms: A Screaming Game for Short-Sighted People.
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FORMULA
| Recurrence: a(n+3) = 7*a(n+2)-9*a(n+1)+a(n)
Generating function: (3-14x+9x^2)/(1-7x+9x^2-x^3)
Formula: a(n) = A^n + B^n + C^n
where, putting u = atan(sqrt(5319)/73), we have:
A = 5.3538557854308282... = (7+2*srqt(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
| a(0) = 3 = A^0+B^0+C^0
a(1) = 7 = A+B+C
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MATHEMATICA
| LinearRecurrence[{7, -9, 1}, {3, 7, 31}, 40] (* From Harvey P. Dale, May 25 2011 *)
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CROSSREFS
| Cf. A141221, A141384.
Sequence in context: A000644 A015459 A115083 * A059296 A123332 A051342
Adjacent sequences: A141382 A141383 A141384 * A141386 A141387 A141388
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KEYWORD
| easy,nice,nonn
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AUTHOR
| Gerard P. Michon (g.michon(AT)att.net), Jul 02 2008, Jul 23 2008
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