cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A008728 Molien series for 3-dimensional group [2,n ] = *22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 217, 224, 231, 238
Offset: 0

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Author

Keywords

Comments

a(n) = A179052(n) for n < 100. - Reinhard Zumkeller, Jun 27 2010

Crossrefs

Programs

  • GAP
    a:=[1,2,3,4,5,6,7,8,9,10,12,14];; for n in [13..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-10]-2*a[n-11]+a[n-12]; od; a; # G. C. Greubel, Jul 30 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^10)) )); // G. C. Greubel, Jul 30 2019
    
  • Maple
    g:= 1/((1-x)^2*(1-x^10)); gser:= series(g, x=0,72); seq(coeff(gser, x, n), n=0..70); # modified by G. C. Greubel, Jul 30 2019
  • Mathematica
    CoefficientList[Series[1/((1-x)^2(1-x^10)), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
  • PARI
    my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^10))) \\ G. C. Greubel, Jul 30 2019
    
  • Sage
    (1/((1-x)^2*(1-x^10))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019
    

Formula

G.f.: 1/((1-x)^2*(1-x^10)).
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+10} floor(j/10).
a(n-10) = (1/2)*floor(n/10)*(2*n - 8 - 10*floor(n/10)). (End)

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A008726 Molien series 1/((1-x)^2*(1-x^8)) for 3-dimensional group [2,n] = *22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168, 175, 182, 189, 196, 203, 210, 217, 224, 232, 240, 248, 256, 264, 272, 280
Offset: 0

Views

Author

Keywords

Crossrefs

Programs

  • GAP
    a:=[1,2,3,4,5,6,7,8,10,12];; for n in [11..80] do a[n]:=2*a[n-1] -a[n-2]+a[n-8]-2*a[n-9]+a[n-10]; od; a; # G. C. Greubel, Sep 09 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)^2*(1-x^8)) )); // G. C. Greubel, Sep 09 2019
    
  • Maple
    seq(coeff(series(1/(1-x)^2/(1-x^8), x, n+1), x, n), n=0..80);
  • Mathematica
    CoefficientList[Series[1/((1-x)^2*(1-x^8)), {x,0,80}], x] (* Vincenzo Librandi, Jun 11 2013 *)
    LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,10,12}, 80] (* Harvey P. Dale, Jan 07 2015 *)
  • PARI
    my(x='x+O('x^80)); Vec(1/((1-x)^2*(1-x^8))) \\ G. C. Greubel, Sep 09 2019
    
  • Sage
    def A008726_list(prec):
        P. = PowerSeriesRing(ZZ, prec)
        return P(1/((1-x)^2*(1-x^8))).list()
    A008726_list(80) # G. C. Greubel, Sep 09 2019
    

Formula

G.f.: 1/((1-x)^2*(1-x^8)).
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+8} floor(j/8).
a(n-8) = (1/2)*floor(n/8)*(2*n-6-8*floor(n/8)). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10). - R. J. Mathar, Apr 20 2010

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010
Minor edits by Jon E. Schoenfield, Mar 28 2014

A008727 Molien series for 3-dimensional group [2,n] = *22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252
Offset: 0

Views

Author

Keywords

Comments

Number of partitions of n into two kinds of 1's and one kind of 9. - Joerg Arndt, Dec 27 2014

Crossrefs

Programs

  • GAP
    a:=[1,2,3,4,5,6,7,8,9,11,13];; for n in [12..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-9]-2*a[n-10]+a[n-11]; od; a; # G. C. Greubel, Sep 09 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^9)) )); // G. C. Greubel, Sep 09 2019
    
  • Maple
    seq(coeff(series(1/((1-x)^2*(1-x^9)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 09 2019
  • Mathematica
    Drop[Accumulate[Floor[Range[70]/9]], 8] (* Jean-François Alcover, Mar 27 2013 *)
    CoefficientList[Series[1/(1-x)^2/(1-x^9), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
    LinearRecurrence[{2,-1,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,11,13},120] (* Harvey P. Dale, Feb 13 2022 *)
  • PARI
    Vec(1/(1-x)^2/(1-x^9)+O(x^66)) /* Joerg Arndt, Mar 27 2013 */
    
  • Sage
    def A008727_list(prec):
        P. = PowerSeriesRing(ZZ, prec)
        return P(1/((1-x)^2*(1-x^9))).list()
    A008727_list(70) # G. C. Greubel, Sep 09 2019
    

Formula

G.f.: 1/((1-x)^2*(1-x^9)).
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+9} floor(j/9).
a(n-9) = (1/2)*floor(n/9)*(2*n - 7 - 9*floor(n/9)). (End)

A011867 a(n) = floor(n*(n-1)/14).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 24, 27, 30, 33, 36, 39, 42, 46, 50, 54, 58, 62, 66, 70, 75, 80, 85, 90, 95, 100, 105, 111, 117, 123, 129, 135, 141, 147, 154, 161, 168, 175, 182, 189, 196, 204, 212, 220, 228, 236, 244, 252, 261, 270, 279, 288, 297
Offset: 0

Views

Author

Keywords

Crossrefs

Programs

  • Magma
    [Floor(n*(n-1)/14): n in [0..70]]; // Vincenzo Librandi, Aug 29 2011
  • Mathematica
    Table[Floor[n(n - 1)/14], {n, 0, 59}] (* Alonso del Arte, Aug 26 2011 *)
    LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,0,0,0,0,1,2,3,4},70] (* Harvey P. Dale, Oct 22 2016 *)

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9).
G.f.: x^5 / ((1-x)^3*(x^6+x^5+x^4+x^3+x^2+x+1) ). (End)

A008730 Molien series 1/((1-x)^2*(1-x^12)) for 3-dimensional group [2,n] = *22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 186, 192, 198, 204
Offset: 0

Views

Author

Keywords

Examples

			..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
550..560..570..580..590..600..610..620..630..640..650..660
...
The columns are: A051866, A139267, A094159, A033579, A049452, A033581, A049453, A033580, A195319, A202804, A211014, A049598
- _Philippe Deléham_, Apr 03 2013
		

Crossrefs

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^12)) )); // G. C. Greubel, Jul 30 2019
    
  • Maple
    seq(coeff(series(1/(1-x)^2/(1-x^12), x, n+1), x, n), n=0..80);
  • Mathematica
    CoefficientList[Series[1/((1-x)^2*(1-x^12)), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
    LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,10,11,12,14,16},70] (* Harvey P. Dale, Jan 01 2024 *)
  • PARI
    my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ G. C. Greubel, Jul 30 2019
    
  • Sage
    (1/((1-x)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

Formula

G.f. 1/( (1-x)^3 * (1+x) *(1+x+x^2) *(1-x+x^2) * (1+x^2) *(1-x^2+x^4)). - R. J. Mathar, Aug 11 2021
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+12} floor(j/12).
a(n-12) = (1/2)*floor(n/12)*(2*n - 10 - 12*floor(n/12)). (End)
a(n) = A221912(n+12). - Philippe Deléham, Apr 03 2013

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010
Showing 1-5 of 5 results.