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%I #70 Sep 01 2023 04:20:39
%S 0,1,7,23,54,105,181,287,428,609,835,1111,1442,1833,2289,2815,3416,
%T 4097,4863,5719,6670,7721,8877,10143,11524,13025,14651,16407,18298,
%U 20329,22505,24831,27312,29953,32759,35735,38886,42217,45733,49439
%N Number of atoms in a decahedron with n shells.
%C 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
%C 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
%C 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
%C a(n) = Sum_{k=1..n} A215630(n,k) for n > 0. - _Reinhard Zumkeller_, Nov 11 2012
%C a(n) - a(n-2) = A010001(n-1), for n>1. - _K. G. Stier_, Dec 21 2012
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A004068/b004068.txt">Table of n, a(n) for n = 0..5000</a>
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, 273 (1996), 199-241, eq. (3).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 5*binomial(n + 1, 3) + binomial(n, 1).
%F a(n) = 5*n^3/6 + n/6.
%F a(n) = Sum_{i=0..n-1} A005891(i). - Xavier Acloque, Oct 08 2003
%F G.f.: x*(1+3*x+x^2) / (1-x)^4. - _R. J. Mathar_, Jun 05 2011
%F E.g.f.: (x/6)*(5x^2 + 15x + 6)*exp(x). - _G. C. Greubel_, Sep 27 2015
%F 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
%t Table[5*n^3/6+n/6,{n,0,80}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 18 2011 *)
%o (Magma) [5*n^3/6+n/6: n in [0..50]]; // _Vincenzo Librandi_, May 15 2011
%o (Maxima) A004068(n):=5*n^3/6+n/6$ makelist(A004068(n),n,0,20); /* _Martin Ettl_, Jan 07 2013 */
%o (PARI) a(n)=5*n^3/6+n/6 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y (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.
%Y Cf. A000292, A005891, A006322 (partial sums), A010001, A081436, A100145, A215630.
%K nonn,easy
%O 0,3
%A Albert D. Rich (Albert_Rich(AT)msn.com)
%E Typo in definition corrected by _Jean M. Morales_, Aug 11 2013
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