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 1-3 of 3 results.

A218470 Partial sums of floor(n/9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 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
Offset: 0

Views

Author

Philippe Deléham, Mar 26 2013

Keywords

Comments

Apart from the initial zeros, the same as A008727.

Examples

			As square array:
..0....0....0....0....0....0....0....0....0....
..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....
...

Links

Crossrefs

Cf. similar sequences: A118729, A174109, A174738.

Programs

Formula

a(9n) = A051682(n).
a(9n+1) = A062708(n).
a(9n+2) = A062741(n).
a(9n+3) = A022266(n).
a(9n+4) = A022267(n).
a(9n+5) = A081266(n).
a(9n+6) = A062725(n).
a(9n+7) = A062728(n).
a(9n+8) = A027468(n).
G.f.: x^9/((1-x)^2*(1-x^9)). - Bruno Berselli, Mar 27 2013

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008732.

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

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

N. J. A. Sloane

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

Links

Crossrefs

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008728, A008729, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010
Cf. A221912

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 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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