A277388 Number of nonnegative solutions of a certain system of linear Diophantine equations depending on an odd parameter.

3, 47, 306, 1270, 4005, 10493, 24052, 49836, 95415, 171435, 292358, 477282, 750841, 1144185, 1696040, 2453848, 3474987, 4828071, 6594330, 8869070, 11763213, 15404917, 19941276, 25540100, 32391775, 40711203, 50739822, 62747706, 77035745, 93937905, 113823568, 137099952, 164214611, 195658015

Offset: 2

Author

Kamil Bradler, Oct 12 2016

Keywords

• nonn
• easy

Comments

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.

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.

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Kamil Bradler, On the number of nonnegative solutions of a system of linear Diophantine equations, arXiv:1610.06387 [math-ph], 2016.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A277387.

Programs

Mathematica

(* 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: *)
SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->1000}];
(* For a very large n the parameter value (1000) must be increased further but the enumeration is increasingly time-consuming. *)
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
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 *)

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

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

Formula

a(n) = n*(-1+n)*(3-2*n+n^2-n^3+2*n^4)/18.

From Colin Barker, Oct 12 2016: (Start)

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.

G.f.: x^2*(3+26*x+40*x^2+10*x^3+x^4) / (1-x)^7.

(End)