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A100538
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Volume of the 3-dimensional box of sides of length equal to consecutive Padovan numbers (A000931). These boxes form a spiral in three dimensions similar to the spiral of Fibonacci boxes in two dimensions.
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3
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1, 2, 4, 12, 24, 60, 140, 315, 756, 1728, 4032, 9408, 21756, 50764, 117845, 273910, 637260, 1480404, 3442800, 8003000, 18603000, 43251975, 100540440, 233735040, 543371136, 1263161472, 2936540824, 6826574552, 15869878969, 36893076570
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OFFSET
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1,2
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COMMENTS
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,2,3,-2,4,-4,-1,-1,0,-1).
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FORMULA
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For large n a(n+1) -> a(n) * p^3 where p is the plastic number = 1.324718... a(n+1) = a(n)+ (a(n)/P(n))*P(n+1 ) where P are the Padovan numbers (A000931) starting 1, 1, 1, 2, 2, 3, 4, 5, 7, etc.
a(n) = +a(n-1) +2*a(n-2) +3*a(n-3) -2*a(n-4) +4*a(n-5) -4*a(n-6) -a(n-7) -a(n-8) -a(n-10) = A000931(n+4)*A000931(n+5)*A000931(n+6). G.f.: x*(1+x+x^3) / ( (x-1)*(x^3-2*x^2+3*x-1)*(x^6+3*x^5+5*x^4+5*x^3+5*x^2+3*x+1) ). - R. J. Mathar, Sep 14 2010
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MATHEMATICA
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LinearRecurrence[{1, 2, 3, -2, 4, -4, -1, -1, 0, -1}, {1, 2, 4, 12, 24, 60, 140, 315, 756, 1728}, 50] (* Vincenzo Librandi, Apr 24 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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