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%I #32 Aug 04 2017 11:53:14
%S 1,17,138,670,2355,6671,16212,35148,69765,129085,225566,375882,601783,
%T 931035,1398440,2046936,2928777,4106793,5655730,7663670,10233531,
%U 13484647,17554428,22600100,28800525,36358101,45500742,56483938,69592895,85144755,103490896,125019312,150157073,179372865,213179610
%N Number of nonnegative solutions of a certain system of linear Diophantine equations depending on an even parameter.
%C The Diophantine system is 2*a_{i,i} + Sum_{j=1..4}*a_{i,j}=n, where i=1..4, j is NOT equal to i and n>=0 is even.
%C It can be proved that the number of nonnegative solutions is e(n) = (2 + n)*(4 + n)*(72 + n*(5 + n)*(12 + n*(4 + n)))/576 and a(n) = n*(1+n)*(3+2*n+n^2+n^3+2*n^4)/18 is obtained by remapping n->2*n-2.
%H Colin Barker, <a href="/A277387/b277387.txt">Table of n, a(n) for n = 1..1000</a>
%H Kamil Bradler, <a href="https://arxiv.org/abs/1610.06387">On the number of nonnegative solutions of a system of linear Diophantine equations</a>, arXiv:1610.06387 [math-ph], 2016.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = n*(1+n)*(3+2*n+n^2+n^3+2*n^4)/18.
%F From _Colin Barker_, Oct 12 2016: (Start)
%F a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) for n>7.
%F G.f.: x*(1+10*x+40*x^2+26*x^3+3*x^4) / (1-x)^7.
%F (End)
%t (* The code is in the InputForm form to simply copy and paste it in Mathematica. The input parameter is n>=0 (even) and for larger n's the code must be preceded by *)
%t SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->1000}];
%t (* For a very large n the parameter value (1000) must be increased further but the enumeration is increasingly time-consuming. *)
%t Reduce[Subscript[a,1,2]+Subscript[a,1,3]+Subscript[a,1,4]==n-2*Subscript[a,1,1]&&Subscript[a,1,2]>=0&&Subscript[a,1,3]>=0&&Subscript[a,1,4]>=0&&Subscript[a,1,1]>=0&&Subscript[a,1,2]+Subscript[a,2,3]+Subscript[a,2,4]==n-2*Subscript[a,2,2]&&Subscript[a,2,3]>=0&&Subscript[a,2,4]>=0&&Subscript[a,2,2]>=0&&Subscript[a,1,3]+Subscript[a,2,3]+Subscript[a,3,4]==n-2*Subscript[a,3,3]&&Subscript[a,3,4]>=0&&Subscript[a,3,3]>=0&&Subscript[a,1,4]+Subscript[a,2,4]+Subscript[a,3,4]==n-2*Subscript[a,4,4]&&Subscript[a,4,4]>=0,{Subscript[a,1,1],Subscript[a,1,2],Subscript[a,1,3],Subscript[a,1,4],Subscript[a,2,2],Subscript[a,2,3],Subscript[a,2,4],Subscript[a,3,3],Subscript[a,3,4],Subscript[a,4,4]},Integers]//Length
%t (*For the special case n=0 the Reduce command must be put in the curly brackets before Length is applied.*)
%o (PARI) a(n) = (18+57*n+86*n^2+81*n^3+47*n^4+15*n^5+2*n^6)/18 \\ _Colin Barker_, Oct 12 2016
%o (PARI) Vec(x*(1+10*x+40*x^2+26*x^3+3*x^4)/(1-x)^7 + O(x^30)) \\ _Colin Barker_, Oct 16 2016
%Y Cf. A277388.
%K nonn,easy
%O 1,2
%A _Kamil Bradler_, Oct 12 2016
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