A100538 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.

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

Offset: 1

Author

John Lien, Nov 27 2004

Keywords

• nonn

Comments

a(n)^(1/3) rounded to the nearest integer equals A000931(n+5). - Peter M. Chema, Apr 24 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• I. Stewart, Tales of a Neglected Number
• Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
• Index entries for linear recurrences with constant coefficients, signature (1,2,3,-2,4,-4,-1,-1,0,-1).

Crossrefs

Cf. A000931.

Programs

Mathematica

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 *)

Formula

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

Extensions

More terms from R. J. Mathar, Sep 14 2010