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A277387 Number of nonnegative solutions of a certain system of linear Diophantine equations depending on an even parameter. 2
1, 17, 138, 670, 2355, 6671, 16212, 35148, 69765, 129085, 225566, 375882, 601783, 931035, 1398440, 2046936, 2928777, 4106793, 5655730, 7663670, 10233531, 13484647, 17554428, 22600100, 28800525, 36358101, 45500742, 56483938, 69592895, 85144755, 103490896, 125019312, 150157073, 179372865, 213179610 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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>=0 is even.

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.

LINKS

Colin Barker, Table of n, a(n) for n = 1..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>7.

G.f.: x*(1+10*x+40*x^2+26*x^3+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>=0 (even) 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

(*For the special case n=0 the Reduce command must be put in the curly brackets before Length is applied.*)

PROG

(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

(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

CROSSREFS

Cf. A277388.

Sequence in context: A085958 A120784 A271395 * A099922 A298626 A157360

Adjacent sequences:  A277384 A277385 A277386 * A277388 A277389 A277390

KEYWORD

nonn,easy

AUTHOR

Kamil Bradler, Oct 12 2016

STATUS

approved

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Last modified August 8 11:31 EDT 2020. Contains 336298 sequences. (Running on oeis4.)