A277388 - OEIS

%I #37 Jun 21 2024 14:45:42

%S 3,47,306,1270,4005,10493,24052,49836,95415,171435,292358,477282,

%T 750841,1144185,1696040,2453848,3474987,4828071,6594330,8869070,

%U 11763213,15404917,19941276,25540100,32391775,40711203,50739822,62747706,77035745,93937905,113823568,137099952,164214611,195658015

%N Number of nonnegative solutions of a certain system of linear Diophantine equations depending on an odd 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>=1 is odd.

%C It can be proved that the number of nonnegative solutions is d(n) = (1 + n)*(3 + n)*(72 + n*(5 + n)*(17 + n*(6 + 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-3.

%H Colin Barker, <a href="/A277388/b277388.txt">Table of n, a(n) for n = 2..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>8.

%F G.f.: x^2*(3+26*x+40*x^2+10*x^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>=1 (odd) 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 Table[(n(n-1)(2n^4-n^3+n^2-2n+3))/18,{n,2,40}] (* or *) Drop[CoefficientList[ Series[ x^2(3+26x+40x^2+10x^3+x^4)/(1-x)^7,{x,0,40}],x],2] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{3,47,306,1270,4005,10493,24052},40] (* _Harvey P. Dale_, Jun 21 2024 *)

%o (PARI) a(n) = (54+189*n+275*n^2+213*n^3+92*n^4+21*n^5+2*n^6)/18 \\ _Colin Barker_, Oct 12 2016

%o (PARI) Vec(x^2*(3+26*x+40*x^2+10*x^3+x^4)/(1-x)^7 + O(x^40)) \\ _Colin Barker_, Oct 16 2016

%Y Cf. A277387.

%K nonn,easy

%O 2,1

%A _Kamil Bradler_, Oct 12 2016