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

2,1

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).

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)

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 *)

Prog

(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

Crossrefs

Cf. A277387.

Sequence in context: A052187 A260219 A131465 * A245014 A247024 A137611

Adjacent sequences: A277385 A277386 A277387 * A277389 A277390 A277391

Keyword

nonn,easy

Author

Kamil Bradler, Oct 12 2016

Status

approved