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A277388 Number of nonnegative solutions of a certain system of linear Diophantine equations depending on an odd parameter. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified August 13 21:57 EDT 2020. Contains 336454 sequences. (Running on oeis4.)