A004068 Number of atoms in a decahedron with n shells.

0, 1, 7, 23, 54, 105, 181, 287, 428, 609, 835, 1111, 1442, 1833, 2289, 2815, 3416, 4097, 4863, 5719, 6670, 7721, 8877, 10143, 11524, 13025, 14651, 16407, 18298, 20329, 22505, 24831, 27312, 29953, 32759, 35735, 38886, 42217, 45733, 49439

Offset: 0

Author

Albert D. Rich (Albert_Rich(AT)msn.com)

Keywords

• nonn
• easy

Comments

Also as a(n)=(n/6)*(5*n^2+1), n>0: structured pentagonal diamond numbers (vertex structure 6) (cf. A081436 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Number of atoms in decahedron with n shells, number = 5/6*(n^3) + 1/6*(n) (T. P. Martin, Shells of atoms, eq.(3)). - Brigitte Stepanov, Jul 02 2011

a(n+1) is the number of triples (w,x,y) having all terms in {0,...,n} and x+y >= w. - Clark Kimberling, Jun 14 2012

a(n) = Sum_{k=1..n} A215630(n,k) for n > 0. - Reinhard Zumkeller, Nov 11 2012

a(n) - a(n-2) = A010001(n-1), for n>1. - K. G. Stier, Dec 21 2012

a(n) is also a figurate number representing a cube of side n with a vertex cut off by a tetrahedron of side n-1. As such, a(n) = A000578(n) - A000292(n-1), n > 0. - Jean M. Morales, Aug 11 2013

The sequence starting with 1 is the third partial sum of (1, 4, 5, 5, 5, ...) and the binomial transform of (1, 6, 10, 5, 0, 0, 0, ...). - Gary W. Adamson, Sep 27 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (3).
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Cf. A000292, A005891, A006322 (partial sums), A010001, A081436, A100145, A215630.

Programs

Magma

[5*n^3/6+n/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

Mathematica

Table[5*n^3/6+n/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

Maxima

A004068(n):=5*n^3/6+n/6$ makelist(A004068(n),n,0,20); /* Martin Ettl, Jan 07 2013 */

PARI

a(n)=5*n^3/6+n/6 \\ Charles R Greathouse IV, Sep 24 2015

Python

def A004068(n): return n*(5*n**2+1)//6 # Chai Wah Wu, Mar 25 2025

Formula

a(n) = 5*binomial(n + 1, 3) + binomial(n, 1).

a(n) = 5*n^3/6 + n/6.

a(n) = Sum_{i=0..n-1} A005891(i). - Xavier Acloque, Oct 08 2003

G.f.: x*(1+3*x+x^2) / (1-x)^4. - R. J. Mathar, Jun 05 2011

E.g.f.: (x/6)*(5x^2 + 15x + 6)*exp(x). - G. C. Greubel, Sep 27 2015

Sum_{n>0} 1/a(n) = 3*(2*gamma + polygamma(0, 1-i/sqrt(5)) + polygamma(0, 1+i/sqrt(5))) = 1.233988011257952852492845364799197179252... where i denotes the imaginary unit. - Stefano Spezia, Aug 31 2023

Extensions

Typo in definition corrected by Jean M. Morales, Aug 11 2013