A071910 a(n) = t(n)*t(n+1)*t(n+2), where t() are the triangular numbers.

0, 18, 180, 900, 3150, 8820, 21168, 45360, 89100, 163350, 283140, 468468, 745290, 1146600, 1713600, 2496960, 3558168, 4970970, 6822900, 9216900, 12273030, 16130268, 20948400, 26910000, 34222500, 43120350, 53867268, 66758580, 82123650, 100328400, 121777920

Offset: 0

Author

N. J. A. Sloane, Jun 13 2002

Keywords

• nonn

Comments

a(n) is also the number of three-dimensional cage assemblies such that the assembly is not a cube. See also A052149 for the two-dimensional version and to A059827 for the non-exclusive version. - Alejandro Rodriguez, Oct 20 2020

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A006542, (first differences of a(n) /18) A006414, (second differences of a(n) /18) A006322, (third differences of a(n) /18) A004068, (fourth differences of a(n) /18) A005891, (fifth differences of a(n) /18) A008706.

Programs

Mathematica

Join[{0},Times@@@Partition[Accumulate[Range[40]],3,1]] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,18,180,900,3150,8820,21168},40] (* Harvey P. Dale, Aug 08 2025 *)

PARI

t(n) = n*(n+1)/2;
a(n) = t(n)*t(n+1)*t(n+2); \\ Michel Marcus, Oct 21 2015

Formula

a(n) = 18*A006542(n+3). - Vladeta Jovovic, Jun 14 2002

G.f.: 18*x*(1+3*x+x^2)/(1-x)^7. - Vladeta Jovovic, Jun 14 2002

a(n) = ((n+1)*(n+2))^3/8 - Sum_{i=1..n+1} i^3. - Jon Perry, Feb 13 2004

a(n) = C(2+n, n)*C(3+n, 1+n)*C(4+n, 2+n). - Zerinvary Lajos, Jul 29 2005

a(n) = A059827(n+1) - A000537(n+1). - Michel Marcus, Oct 21 2015