A008729 - OEIS

%I #44 Sep 08 2022 08:44:36

%S 1,2,3,4,5,6,7,8,9,10,11,13,15,17,19,21,23,25,27,29,31,33,36,39,42,45,

%T 48,51,54,57,60,63,66,70,74,78,82,86,90,94,98,102,106,110,115,120,125,

%U 130,135,140,145,150,155,160,165,171,177,183,189,195,201,207,213,219

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

%H Vincenzo Librandi, <a href="/A008729/b008729.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=194">Encyclopedia of Combinatorial Structures 194</a>

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,0,0,0,1,-2,1).

%F From _Mitch Harris_, Sep 08 2008: (Start)

%F a(n) = Sum_{j=0..n+11} floor(j/11).

%F a(n-11) = (1/2)*floor(n/11)*(2*n - 9 - 11*floor(n/11)). (End)

%F a(n) = A218530(n+11). - _Philippe Deléham_, Apr 03 2013

%F From _Chai Wah Wu_, Jul 08 2016: (Start)

%F a(n) = 2*a(n-1) - a(n-2) + a(n-11) - 2*a(n-12) + a(n-13) for n > 12.

%F G.f.: 1/(1 - 2*x + x^2 - x^11 + 2*x^12 - x^13) = 1/((1-x)^3 *(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)). (End)

%e ..1....2....3....4....5....6....7....8....9...10...11

%e .13...15...17...19...21...23...25...27...29...31...33

%e .36...39...42...45...48...51...54...57...60...63...66

%e .70...74...78...82...86...90...94...98..102..106..110

%e 115..120..125..130..135..140..145..150..155..160..165

%e 171..177..183..189..195..201..207..213..219..225..231

%e 238..245..252..259..266..273..280..287..294..301..308

%e 316..324..332..340..348..356..364..372..380..388..396

%e 405..414..423..432..441..450..459..468..477..486..495

%e 505..515..525..535..545..555..565..575..585..595..605

%e ...

%e The first six columns are A051865, A180223, A022268, A022269, A211013, A152740.

%e - _Philippe Deléham_, Apr 03 2013

%p g:= 1/((1-x)^2*(1-x^11)); gser:= series(g, x=0,72); seq(coeff(gser, x, n), n=0..70); # modified by _G. C. Greubel_, Jul 30 2019

%t CoefficientList[Series[1/((1-x)^2*(1-x^11)), {x,0,70}], x] (* _Vincenzo Librandi_, Jun 11 2013 *)

%o (PARI) my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^11))) \\ _G. C. Greubel_, Jul 30 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^11)) )); // _G. C. Greubel_, Jul 30 2019

%o (Sage) (1/((1-x)^2*(1-x^11))).series(x, 70).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 30 2019

%o (GAP) a:=[1,2,3,4,5,6,7,8,9,10,11,13,15];; for n in [14..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-11]-2*a[n-12]+a[n-13]; od; a; # _G. C. Greubel_, Jul 30 2019

%Y Cf. A001840, A001972, A008724-A008728, A008732, A218530.

%K nonn,tabf,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Vladimir Joseph Stephan Orlovsky_, Mar 14 2010