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%I #30 Sep 08 2022 08:44:36
%S 1,0,2,1,3,2,5,3,7,5,9,7,12,9,15,12,18,15,22,18,26,22,30,26,35,30,40,
%T 35,45,40,51,45,57,51,63,57,70,63,77,70,84,77,92,84,100,92,108,100,
%U 117,108,126,117,135,126,145,135,155,145,165,155,176,165,187
%N Molien series for 3-dimensional group [2, n] = *22n.
%C a(n+4) is the number of solutions to the equation X + Y + Z = n such that X < Z, Y < Z, and X + Y >= Z. - _Geoffrey Critzer_, Jul 13 2013
%C Number of partitions of n into two sorts of 2, and one sort of 3. - _Joerg Arndt_, Jul 14 2013
%H G. C. Greubel, <a href="/A008731/b008731.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=222">Encyclopedia of Combinatorial Structures 222</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,-1,-2,0,1).
%F G.f.: 1/((1-x^2)^2*(1-x^3)) = 1/((1-x)^3*(1+x)^2*(1+x+x^2)).
%F a(n) = (1/48)*(2*n^2 + 14*n + 27 + (6*n+21)*(-1)^n - 16(n=1 mod 3)).
%F Euler transform of length 3 sequence [ 0, 2, 1]. - _Michael Somos_, Feb 02 2015
%F a(n) = a(-7-n) for all n in Z. - _Michael Somos_, Feb 02 2015
%F 0 = a(n) + a(n+1) - a(n+2) - 2*a(n+3) - a(n+4) + a(n+5) + a(n+6) - 1 for all n in Z. - _Michael Somos_, Feb 02 2015
%F a(n+3) - a(n) = 0 if n even, (n+5)/2 otherwise. - _Michael Somos_, Feb 02 2015
%F |a(n)-a(n-1)| = A154958(n). - _R. J. Mathar_, Aug 11 2021
%e a(4) = 3 because we have:
%e 1 + 3 + 4 = 2 + 2 + 4 = 3 + 1 + 4. - _Geoffrey Critzer_, Jul 13 2013
%e G.f. = 1 + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 5*x^6 + 3*x^7 + 7*x^8 + 5*x^9 + ...
%p seq(coeff(series(1/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0 .. 70); # modified by _G. C. Greubel_, Jul 30 2019
%t CoefficientList[Series[1/(1-x^2)^2/(1-x^3),{x,0,70}],x] (* _Geoffrey Critzer_, Jul 13 2013 *)
%t a[ n_] := Quotient[ (2 n^2 + If[ OddQ[n], 8 n + 6, 20 n + 48]), 70]; (* _Michael Somos_, Feb 02 2015 *)
%t a[ n_] := Module[{m=n}, If[ n < 0, m=-7-n]; SeriesCoefficient[ 1 / ( (1 - x^2)^2 * (1 - x^3)), {x, 0, m}]]; (* _Michael Somos_, Feb 02 2015 *)
%t LinearRecurrence[{0,2,1,-1,-2,0,1},{1,0,2,1,3,2,5},70] (* _Harvey P. Dale_, Feb 23 2018 *)
%o (PARI) {a(n) = (2*n^2 + if( n%2, 8*n + 6, 20*n + 48)) \ 48}; /* _Michael Somos_, Feb 02 2015 */
%o (PARI) {a(n) = if( n<0, n=-7-n); polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3)) + x * O(x^n), n)}; /* _Michael Somos_, Feb 02 2015 */
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x^2)^2*(1-x^3)) )); // _G. C. Greubel_, Jul 30 2019
%o (Sage) (1/((1-x^2)^2*(1-x^3))).series(x, 70).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 30 2019
%o (GAP) a:=[1,0,2,1,3,2,5];; for n in [8..70] do a[n]:=2*a[n-2]+a[n-3]-a[n-4]-2*a[n-5]+a[n-7]; od; a; # _G. C. Greubel_, Jul 30 2019
%Y First differences of A008763.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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