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%I #46 Jan 01 2024 13:09:37
%S 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,
%T 45,48,51,54,57,60,63,66,69,72,76,80,84,88,92,96,100,104,108,112,116,
%U 120,125,130,135,140,145,150,155,160,165,170,175,180,186,192,198,204
%N Molien series 1/((1-x)^2*(1-x^12)) for 3-dimensional group [2,n] = *22n.
%H Vincenzo Librandi, <a href="/A008730/b008730.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=195">Encyclopedia of Combinatorial Structures 195</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).
%F 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
%F From _Mitch Harris_, Sep 08 2008: (Start)
%F a(n) = Sum_{j=0..n+12} floor(j/12).
%F a(n-12) = (1/2)*floor(n/12)*(2*n - 10 - 12*floor(n/12)). (End)
%F a(n) = A221912(n+12). - _Philippe Deléham_, Apr 03 2013
%e ..1....2....3....4....5....6....7....8....9...10...11...12
%e .14...16...18...20...22...24...26...28...30...32...34...36
%e .39...42...45...48...51...54...57...60...63...66...69...72
%e .76...80...84...88...92...96..100..104..108..112..116..120
%e 125..130..135..140..145..150..155..160..165..170..175..180
%e 186..192..198..204..210..216..222..228..234..240..246..252
%e 259..266..273..280..287..294..301..308..315..322..329..336
%e 344..352..360..368..376..384..392..400..408..416..424..432
%e 441..450..459..468..477..486..495..504..513..522..531..540
%e 550..560..570..580..590..600..610..620..630..640..650..660
%e ...
%e The columns are: A051866, A139267, A094159, A033579, A049452, A033581, A049453, A033580, A195319, A202804, A211014, A049598
%e - _Philippe Deléham_, Apr 03 2013
%p seq(coeff(series(1/(1-x)^2/(1-x^12), x, n+1), x, n), n=0..80);
%t CoefficientList[Series[1/((1-x)^2*(1-x^12)), {x,0,70}], x] (* _Vincenzo Librandi_, Jun 11 2013 *)
%t 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 *)
%o (PARI) my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ _G. C. Greubel_, Jul 30 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^12)) )); // _G. C. Greubel_, Jul 30 2019
%o (Sage) (1/((1-x)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 30 2019
%Y Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008728, A008729, A008732. - _Vladimir Joseph Stephan Orlovsky_, Mar 14 2010
%Y Cf. A221912
%K nonn,tabf,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladimir Joseph Stephan Orlovsky_, Mar 14 2010
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