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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
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.
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: A357550 A271395 A358202 * A099922 A298626 A157360
KEYWORD
nonn,easy
AUTHOR
Kamil Bradler, Oct 12 2016
STATUS
approved