%I #29 Oct 28 2022 14:22:53

%S 1,3,6,12,21,33,50,72,99,133,174,222,279,345,420,506,603,711,832,966,

%T 1113,1275,1452,1644,1853,2079,2322,2584,2865,3165,3486,3828,4191,

%U 4577,4986,5418,5875,6357,6864,7398,7959,8547,9164,9810,10485

%N Expansion of (1 + x)*(1 - x + x^2)/((1 - x)^4*(1 + x + x^2)).

%C a(n) is the number of 3 X 3 matrices with nonnegative integer entries such that every row sum, column sum and the trace of the matrix is n. - Sharon Sela (sharonsela(AT)hotmail.com), May 20 2002

%H Vincenzo Librandi, <a href="/A070333/b070333.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).

%F a(n) = 1 + 7*n/6 + n^2/2 + n^3/9 + 2*A049347(n-1)/9. - _R. J. Mathar_, Dec 03 2010

%F From _Mircea Merca_, Dec 03 2010: (Start)

%F a(n) = round((2*n + 3)*(n^2 + 3*n + 6)/18).

%F a(n) = floor((n + 2)*(2*n^2 + 5*n + 11)/18).

%F a(n) = ceiling((n + 1)*(2*n^2 + 7*n + 14)/18).

%F a(n) = round((n + 1)*(2*n^2 + 7*n + 14)/18).

%F a(n) = a(n-3) + n^2 + 2 for n > 2. (End)

%F E.g.f.: exp(x)*(1 + x*(32 + x*(15 + 2*x))/18) + 4*exp(-x/2)*sin(sqrt(3)*x/2)/(9*sqrt(3)). - _Stefano Spezia_, Oct 28 2022

%p A049347 := proc(n) op(1+(n mod 3),[1,-1,0]) ; end proc:

%p A070333 := proc(n) 1+7*n/6+n^2/2+n^3/9+2*A049347(n-1)/9 ; end proc: # _R. J. Mathar_, Dec 03 2010

%t CoefficientList[ Series[(1 + x^3)/(1 - 3*x + 3*x^2 - 2*x^3 + 3*x^4 - 3*x^5 + x^6), {x, 0, 45}], x]

%o (Magma) [Round((2*n+3)*(n^2+3*n+6)/18): n in [0..50]]; // _Vincenzo Librandi_, Jun 25 2011

%o (PARI) Vec((1+x)*(1-x+x^2)/((1-x)^4*(1+x+x^2))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012

%Y Cf. A059827, A001844, A002817.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, May 11 2002