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

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

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 195

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

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

Example

..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

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 *)

Prog

(PARI) my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ G. C. Greubel, Jul 30 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 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

Crossrefs

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

Cf. A221912

Sequence in context: A032962 A032519 A221912 * A033064 A246074 A130225

Adjacent sequences: A008727 A008728 A008729 * A008731 A008732 A008733

Keyword

nonn,tabf,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Status

approved