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

1,2

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).

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

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 *)

Crossrefs

Cf. A000931.

Sequence in context: A057422 A036045 A331392 * A303794 A135139 A161894

Adjacent sequences: A100535 A100536 A100537 * A100539 A100540 A100541

Keyword

nonn

Author

John Lien, Nov 27 2004

Extensions

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

Status

approved