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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Kamil Bradler

Kamil Bradler's wiki page.

Kamil Bradler has authored 2 sequences.

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

Original entry on oeis.org

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

Views

Author

Kamil Bradler, Oct 12 2016

Keywords

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

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)

A277387 Number of nonnegative solutions of a certain system of linear Diophantine equations depending on an even parameter.

Original entry on oeis.org

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

Views

Author

Kamil Bradler, Oct 12 2016

Keywords

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

Crossrefs

Cf. A277388.

Programs

  • 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.*)
  • 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

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)

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