%I #27 Sep 08 2022 08:44:58

%S 1,4,10,21,39,66,104,155,221,304,406,529,675,846,1044,1271,1529,1820,

%T 2146,2509,2911,3354,3840,4371,4949,5576,6254,6985,7771,8614,9516,

%U 10479,11505,12596,13754,14981,16279,17650,19096,20619,22221

%N a(n) = (2*n-1)*(n^2 -n +6)/6.

%H Harvey P. Dale, <a href="/A049480/b049480.txt">Table of n, a(n) for n = 1..1000</a>

%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Rep., 273 (1996), 199-241, eq. (10).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F From _Harvey P. Dale_, Jan 01 2012: (Start)

%F G.f.: x*(x^3 + 1)/(x-1)^4.

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4); a(1)=1, a(2)=4, a(3)=10, a(4)=21. (End)

%F E.g.f.: (-6 + 12*x + 3*x^2 + 2*x^3)*exp(x)/6 + 1. - _G. C. Greubel_, Dec 01 2017

%t Table[(2n-1)(n^2-n+6)/6,{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,4,10,21},50] (* _Harvey P. Dale_, Jan 01 2012 *)

%o (PARI) a(n)=(2*n-1)*(n^2-n+6)/6 \\ _Charles R Greathouse IV_, Sep 24 2015

%o (Magma) [(2*n-1)*(n^2-n+6)/6: n in [1..30]]; // _G. C. Greubel_, Dec 01 2017

%o (PARI) x='x+O('x^30); Vec(serlaplace((-6 + 12*x + 3*x^2 + 2*x^3)*exp(x)/6 + 1)) \\ _G. C. Greubel_, Dec 01 2017

%Y 1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, Aug 01 2001